I think this grim little short story is by del Rey, but I'm not sure.
"'The Years Draw Nigh'", a short story by Lester del Rey. The title is from Ecclesiastes 12:1.
I'm pretty sure it was published in the 1940s/50s.
It was first published in Astounding Science Fiction, October 1951, available at the Internet Archive.
Earth's very last starship is coming home after searching the galaxy for life.
Zeke snorted. He'd spent time enough at Marsport, first as head of communications, and finally as director of the whole Star Ship project, while they built the great ships and sent them out as fast as they could come off the ways. Forty ships during half a century, each costing over four billion dollars. And the Thirty-four was the last one out. All the rest had come back to report failure in this final quest for new frontiers.
Exploration began with high hopes due to the discovery of a Martian civilization which, for reasons unknown, died out.
Then his eyes went to the horizon, where the piled stones and pitted pylon of beryl steel still stood, marking what had been the unknown and apparently unknowable race of Mars, dead perhaps ten million years before. Once that race must have spread its structures across the whole planet, but now there were only such traces as this, useless to even the archaeologists. All the elaborate designs on them might have held significance once, but no man would ever decode them. There was no hint as to their nature, or where the race had vanished—or why.
The crew of this last starship, however, has discovered what the Martians discovered; life is a freak that exists on only Earth and Mars - there is no life anywhere else in the Universe.
"Evidence of life anywhere?" he asked reluctantly, as the other finished. But the question had to be asked, although the answer could be predicted, almost certainly. Even over that distance, the possibility of other races to study might drive the scientists to set up an outpost, and with that as a basis, another world might be developed as a stepping stone to still further exploration.
Miffen's voice was hesitant as the answer came. "The world we were on—Outpost, we called it—had some ruins that could only come from intelligence. But there was nothing living there. Maybe it had been what we called it once—Damn!"
The Martian race lost its will to live after that,
Zeke looked up at the four men, and then out toward the pylon again—all that was left of a race that had searched the stars in its need to find new frontiers. It must have been a hardy race, since it had dared to set up a colony across all those innumerable parsecs of space, without even the inspiration of other life. Then, when that colony failed, the race had returned to the loneliness of its own little world, where the stars looked down grimly, no longer promising anything. Now Mars had been dead ten million years, and the pylon stood as the final tombstone on the world which had become a prison. The old puzzle of that race's end was solved.
and now it seems the same is going to happen to humanity.
The speaker was sputtering with Stendal's impatient questions, as Zeke and the men studied each other, but they gave no attention to it. Preacher Hook sighed, breaking the silence.
"Man goeth to his long home," he quoted softly. "And the mourners go about the streets; or ever the silver cord be loosed, or the golden bowl be broken, or the pitcher be broken at the fountain, or the wheel broken at the cistern; and the dust return to the earth as it was, and the spirit return unto God who gave it."
Zeke nodded and picked up the microphone.
"Just get the whisky. We've decided to skip the rejuvenation."
He put the microphone back on its hook carefully and headed toward the handrails that led down, with the others behind him. Ozin had the rocket waiting, and they climbed in and strapped themselves down.
Then the rockets blasted, and the last five men beyond the Earth were heading home.