Clement of Rome
It may be helpful to read here the words of a man who was taught by Paul, Clement of Rome:
There was Peter who by reason of unrighteous jealousy endured not one not one but many labors, and thus having borne his testimony went
to his appointed place of glory.
By reason of jealousy and strife Paul by his example pointed out the prize of patient endurance. After that he had been seven times in
bonds, had been driven into exile, had been stoned, had preached in
the East and in the West, he won the noble renown which was the reward
of his faith,
having taught righteousness unto the whole world and having reached the farthest bounds of the West; and when he had borne his testimony
before the rulers, so he departed from the world and went unto the
holy place, having been found a notable pattern of patient endurance.
(1 Clement 5:4-6)
Here we have an apostolic father who is suggesting that although Peter & Paul have not yet been resurrected, they are already in a better holier place and have already received some form of glory. Clearly Clement believes there is something between death and the resurrection.
The Spirit isn't going nowhere, it's going somewhere
This would make for a very straightforward interpretation of these passages:
…Stephen, calling upon God, and saying, Lord Jesus, receive my spirit.
And when Jesus had cried with a loud voice, he said, Father, into thy
hands I commend my spirit: and having said thus, he gave up the ghost.
They knew their body was dying but their concern was for their spirit - because they believed their spirit was going to the realm of God.
This is supported by Ecclesiastes 12:7
Then shall the dust return to the earth as it was: and the spirit
shall return unto God who gave it.
Conscious activity by the dead--possibly in the presence of God?
I've offered more in-depth thoughts on Revelation 6:9-11 here making the case that the spirit remains conscious after death; I think these verses are very relevant:
9 And when he had opened the fifth seal, I saw under the altar the
souls of them that were slain for the word of God, and for the
testimony which they held:
10 And they cried with a loud voice, saying, How long, O Lord, holy
and true, dost thou not judge and avenge our blood on them that dwell
on the earth?
11 And white robes were given unto every one of them; and it was said
unto them, that they should rest yet for a little season, until their
fellowservants also and their brethren, that should be killed as they
were, should be fulfilled.
The dead here are speaking, being spoken to, and are being given robes. This suggests that not only are they conscious, but their disembodied spirits have substance.
Whether these passages speak of entering the direct presence of God or entering a spiritual place he has prepared--free from the whims of flawed mortals--perhaps we cannot be entirely certain.
Jesus, Stephen, and Paul all believed that their spirit was going somewhere. Maybe these passages cannot plot that location on a map, but Paul's own disciple Clement reaffirms Paul's understanding: it was not the realm of men, but the realm of God.