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1. [The Inscriptions][ti]
   This category includes texts from diverse geographical sites and spanning a long period of time; however, the total amount of text is limited. As of 2009, the text of the Inscriptions amounted to about 2% of the size of the Hebrew Bible.¹

2. [Ben Sira][bs]
   Ben Sira was 2nd C. BCE Jewish scribe best known for the Wisdom of Sirach (=Ecclesiasticus). That book was only known in its Greek version until manuscripts (dating from the 11th-12th C.) were found in the [Cairo geniza][cg]. Portions have also been found among the DSS. Together, the available text of Hebrew Ben Sira amounts to about 3% of the size of the Hebrew Bible.

3. Dead Sea Scrolls (DSS)
   Our tag wiki provides a helpful introduction. Although, as pointed out there, about 80% of the manuscripts included in this group are non-Biblical, these are, for the most part, considerably shorter. Compiled, they amount to about 20% of the text of the Hebrew Bible.

1. [The Inscriptions][ti]
   This category includes texts from diverse geographical sites and spanning a long period of time; however, the total amount of text is limited. As of 2009, the text of the Inscriptions amounted to about 2% of the size of the Hebrew Bible.¹

2. [Ben Sira][bs]
   Ben Sira was 2nd C. BCE Jewish scribe best known for the Wisdom of Sirach (=Ecclesiasticus). That book was only known in its Greek version until manuscripts (dating from the 11th-12th C.) were found in the [Cairo geniza][cg]. Portions have also been found among the DSS. Together, the available text of Hebrew Ben Sira amounts to about 3% of the size of the Hebrew Bible.

3. Dead Sea Scrolls (DSS)
   Our tag wiki provides a helpful introduction. Although, as pointed out there, about 80% of the manuscripts included in this group are non-Biblical, these are, for the most part, considerably shorter. Compiled, they amount to about 20% of the text of the Hebrew Bible.

The OP asks specifically about extra-biblical resources available for exploring the meaning of rare words in the Bible. The lexical situation is very different when considering Classical Hebrew and Koine Greek, so I will consider those separately. A few general remarks:

1. The best resource for English readers to understand the words of the Bible is the English translations. The OP mentions concern about settling for "what translators felt it to mean". It is admirable to aspire to more in-depth analysis, and this can be an edifying study in itself, but the translators of modern texts have at their disposal many years of education and experience in the languages as well as the best available lexical resources (see below). Any conclusions from one's own word-study that contradict the consensus of translators should be be met with a healthy skepticism.

2. The next best resource for the English speaking student (and, indeed, for all but the most skilled Hebrew and Greek scholars) to determine meaning is the lexicons. It is the job of the lexicographer to peruse the available literature and summarize their findings. Because most lexicons of ancient Greek and Hebrew to some extent also acts as concordances (i.e. they cite examples from period literature), these are included below. We also have [a Meta post with a guide to some of the online resources][meta] available in this area. As I compose this answer, I see we have [another][dan] that reviews the available Greek lexicons.

Hebrew

I think you may be overestimating the available extra-biblical corpus of ancient Hebrew. To my understanding, we have:

1. [The Inscriptions][ti]
   This category includes texts from a variety of geographical sites and spanning a long period of time; however, the total amount of text is limited. As of 2009¹, the text of the Inscriptions amounted to about 2% of the size of the Hebrew Bible.

2. [Ben Sira][bs]
   Ben Sira was 2nd C. BCE Jewish scribe best known for the Wisdom of Sirach (=Ecclesiasticus). That book was only known in its Greek version until manuscripts (dating from the 11th-12th C.) were found in the [Cairo geniza][cg]. Portions have also been found among the DSS. Together, the available text of Hebrew Ben Sira amounts to about 3% of the size of the Hebrew Bible.

3. Dead Sea Scrolls (DSS)
   Our tag wiki provides a helpful introduction. Although, as pointed out there, only about one-fifth of the manuscripts included in this group are Biblical texts, the non-biblical texts are, for the most part, considerably shorter, amounting, when compiled, to about 20% of the text of the Hebrew Bible.

As you can sea, the Hebrew Bible itself remains by far the most expansive source of Classical Hebrew text. The lexical resources in the [Meta post][meta] provide information about this usage. To my knowledge, the only lexicon that also includes the extra-biblical sources (up to ~200 CE) is the [Dictionary of Classical Hebrew][dch], ed. David Clines. From the introduction:

This formidable 8-volume work not only considers extra-biblical sources, it also comprehensively catalogues them in a [syntagmatic analysis][sa] (i.e. listing all the subjects and objects that are attested for every verb, and,for nouns, all the verbs and all the other nouns with which they are connected). Unfortunately, to my knowledge this has no freely available version unless you have access through a library.

For instance, consider [ἅλωσις][alosis] in 2 peter 2:12. Neither this term nor its cognate verb [ἁλίσκομαι][aliskomai] occur elsewhere in the NT. However, you can see that both are well attested in other literature. In the online LSJ linked, many examples are given with active links to the Greek text. The LSJ entries provide representative examples but are not necessarily comprehensive.

[alosisTLG]: http://stephanus.tlg.uci.edu/Iris/demo/tsearch.jsp#qexact=true&ql=A(%2FLWSIS&q=ἅλωσις%2C%20-εως%2C%20ἡ&count=2&dispg=20&query_type=popupfresh&search_type=lemma&search_cat=simple&line_citation=3&td=greek&sortby=author [alosis]: http://stephanus.tlg.uci.edu/lsj/#eid=5073&context=lsj&action=from-search [aliskomai]: http://stephanus.tlg.uci.edu/lsj/#eid=4491&context=lsj&action=from-search

The OP asks specifically about extra-biblical resources available for exploring the meaning of rare words in the Bible. The lexical situation is very different when considering Classical Hebrew and Koine Greek, so I will consider those separately. First, a few general remarks about Biblical word studies:

1. The best resource for English readers to understand the words of the Bible is the English translations. The OP mentions concern about settling for "what translators felt it to mean". It is admirable to aspire to more in-depth analysis, and the investigation itself can be edifying study, but the translators of modern texts have at their disposal many years of education and experience with the languages as well as the best available lexical resources (see below). Any conclusions from one's own word-study that contradict the consensus of translators should be be met with a healthy skepticism.

2. The next best resource for English readers (and, indeed, for all but the most skilled Hebrew and Greek scholars) to determine word meaning is the lexicons. It is the job of the lexicographer to peruse the available literature and summarize their findings. Because most lexicons of ancient Greek and Hebrew to some extent also acts as concordances (i.e. they cite examples from period literature), these are included below. We also have [a Meta post with a guide to some of the online resources][meta] available in this area. As I compose this answer, I see we have [another][dan] that reviews the available Greek lexicons.

##Hebrew

I think you may be overestimating the available extra-biblical corpus of ancient Hebrew. To my understanding, we have:

1. [The Inscriptions][ti]
   This category includes texts from diverse geographical sites and spanning a long period of time; however, the total amount of text is limited. As of 2009, the text of the Inscriptions amounted to about 2% of the size of the Hebrew Bible.¹

2. [Ben Sira][bs]
   Ben Sira was 2nd C. BCE Jewish scribe best known for the Wisdom of Sirach (=Ecclesiasticus). That book was only known in its Greek version until manuscripts (dating from the 11th-12th C.) were found in the [Cairo geniza][cg]. Portions have also been found among the DSS. Together, the available text of Hebrew Ben Sira amounts to about 3% of the size of the Hebrew Bible.

3. Dead Sea Scrolls (DSS)
   Our tag wiki provides a helpful introduction. Although, as pointed out there, about 80% of the manuscripts included in this group are non-Biblical, these are, for the most part, considerably shorter. Compiled, they amount to about 20% of the text of the Hebrew Bible.

As you can sea, the Hebrew Bible itself remains by far most plentiful source of Classical Hebrew text. The lexical resources in the [Meta post][meta] provide information about this usage. To my knowledge, the only lexicon that also includes the extra-biblical sources (up to ~200 CE) is the [Dictionary of Classical Hebrew][dch], ed. David Clines. From the introduction:

This formidable 8-volume work not only considers extra-biblical sources, it also comprehensively catalogues them in a [syntagmatic analysis][sa] (i.e. listing all the subjects and objects that are attested for every verb, and, for nouns, all the verbs and all the other nouns with which they are connected). Unfortunately, to my knowledge this has no freely available version unless you have access through a library.

As an example, consider [ἅλωσις][alosis] in 2 Peter 2:12. Neither this term nor its cognate verb [ἁλίσκομαι][aliskomai] occur elsewhere in the NT. However, you can see that both are well attested in other literature. In the online [Liddell-Scott-Jones Greek-English Lexicon][lsj] (LSJ), many examples are given with active links to the Greek text. The LSJ entries provide representative examples but are not necessarily comprehensive.

[alosisTLG]: http://stephanus.tlg.uci.edu/Iris/demo/tsearch.jsp#qexact=true&ql=A(%2FLWSIS&q=ἅλωσις%2C%20-εως%2C%20ἡ&count=2&dispg=20&query_type=popupfresh&search_type=lemma&search_cat=simple&line_citation=3&td=greek&sortby=author [alosis]: http://stephanus.tlg.uci.edu/lsj/#eid=5073&context=lsj&action=from-search [aliskomai]: http://stephanus.tlg.uci.edu/lsj/#eid=4491&context=lsj&action=from-search [lsj]: http://www.tlg.uci.edu/lsj/