They are literal days. This vision concerns the rise of Antiochus IV Epiphianes and the Maccabean Revolt during the 2nd cent. BC. The time period is pin pointed by verses 20 - 21.
"20 The ram which thou sawest having two horns are the kings of Media and Persia. 21 And the rough goat is the king of Grecia: and the great horn that is between his eyes is the first king." KJV
We can then apply this to the previous vs. 3-14 and know that the ram with two horns (two kings) was Mede and Persian empire, with the greater horn as Cyrus and the lesser horn as Darius (Gubaru) the Mede. (2) (3)
The vision of the goat with the strong horn (vs. 5) was Alexander the Great who conquered the Mede-Persian empire, and became the 3rd empire of Daniel's vision in chap. 2, and chap. 11. Dan. 8:8 tells of the death of Alexander: "when he was strong, the great horn was broken", and the division of the Greek empire into four smaller ones.
The four generals who split Alexander's kingdom are known from history as Seleucus 1 Nicator (the Seleucid dynasty), Ptolemy 1 Soter (the Ptolemaic dynasty), Cassander, and Lysimachus. They were split with the four directions - the four winds of heaven: east, west, north, and south. (4)
Of these four lesser kingdoms one took over the south and east, toward the pleasant land (vs. 9, Israel), and was the Seleucid dynasty in the Eastern Mediterranean regions sometimes called the Levant ("the rising" for the rising sun).
The Seleucid dynasty is referred to as the kings of the north in Dan. ch. 11, which calls the Ptolemaic dynasty the kings of the south. The kings of the north were constantly battling with the kings of the south for control of the Levant / Palestine. (5).
They were north and south of Jerusalem, which is the focus of the prophesies of Daniel. Daniel was praying in chap. 9 about when God would allow them to return to Jerusalem, and Jerusalem was the pleasant land (vs. 9), the holy land, the holy city (Dan. 9:24). So, the directions of north and south were as related to the city of Jerusalem.
The host of heaven in vs. 10 were the people of heaven - God's people which were at that time still the Israelite, and were Daniel's people (Dan. 9:24). Casting some of the host (people) and the stars (rulers, princes) to the ground happened when the prince that magnified himself (vs. 11) to the prince (ruler) of the host (God's people) and took away the daily sacrifice.
The use of "stars" in prophesy meant the rulers, authorities of the land. Throwing the "stars" down was removing them from their positions of rule and authority.
The prince that magnified himself was Antiochus IV Epiphianes whom the Jews called Antiochus Epimanes - the mad man. (7) He changed the succession of the high priest from the father/son line to one of client privilege - bought and paid for. First, Jason (Yeshua) ousted his brother Onias III by buying the position form Antiochus in 175 BC, and then in 172 - 171 BC Antiochus Epiphanes became angry with Jason, and displaced him with Jason's younger brother Onias who changed his name to the Greek Menelaus. Menelaus had outbid Jason for the post.
The 2300 days are literal days, approx. 6 years and 4 months, and are the time from which Antiochus IV Epiphianes sacked Jerusalem, and plundered the temple. From 170 -169 BC to the time that Judah, son of Mattathias (Maccabean) restored and cleansed the temple in approx. Dec 165(4) BC was the time the temple and the host (people) were trodden under foot (vs. 13-14). (1) (6) (8) (9)
1) The Maccabean Revolt - here
2) Daniel's Prophesy of Antiochus Epiphanes - here
3) Belshazzar & Darius the Mede - here
4) Kingdoms of the Successors of Alexander - here
" The kingdom of Cassander (circa 358–297 BC), consisted of Macedonia, most of Greece, and parts of Thrace. The kingdom of Lysimachus (circa 361–281 BC), included Lydia, Ionia, Phrygia, and other parts of present-day Turkey. The kingdom of Seleucus (died 281 BC; later the Seleucid Empire), comprised present-day Iran, Iraq, Syria, and parts of Central Asia. The kingdom of Ptolemy I (died 283 BC) included Egypt and neighboring regions."
5) Pallestine in the Hellenistic Age - here
6) Chronology of Seleucid Rulers and Their Impact on Jerusalem - here
7) Antiochus Epiphianes - here
8) Antiochus Epiphanes - here
9) The Prophesy of Daniel 8 - ApologeticsPress