The main character was called Stone, or Commander Stone. I recall them being in outer space, probably in orbit around Earth. I think their might have been some kind of spider involved, like a space alien maybe. There was some mishap with the ship, and I seem to recall something about like a broken needle in someone's arm, which put them into deep sleep. I do not remember the title, but can only see the off color white book, in paperback, which was not too big.
Trapped in Space by Jack Williamson
Protagonist Jeff Stone travels to the newly discovered star Topaz to find out what happened to his brother Ben. He discovered there were no planets around Topaz, just a huge asteroid belt. The native aliens were large spider-like creatures. Jeff eventually finds his brother and the rest of his crew. They have all taken deep-sleep shots, a drug that puts the users into suspended animation to conserve oxygen. Ben's deep sleep hypo broke off the needle in his arm.
The book can be borrowed from the Internet Archive here. It was published in 1968, within your date range.
There are actually two species of spider-like aliens involved. Early in the book on page 21, we meet "Buzz Dozen-Dozen". He is a friendly telepathic alien about the size of a dog, but with many legs and described as looking "a little like a blown-up spider". The book has pencil illustrations and there is a picture of Buzz on page 23. He is the adopted "brother-sister" of Lupe Flor, a fellow member of the rescue crew.
The second species of spider-like aliens are the much bigger vacuum-dwelling "Rock Hoppers". They are described on page 114 as having five arms that
...were bright as silver, but the body was black - black, round, flat and with no head.
The Rock Hoppers weave vast webs in space, one of which captured Ben's ship. Jeff ends up fighting one of them in space, but when it is at his mercy he finds he cannot bring himself to kill it. In turn it rescues him. With the help of Buzz, who has similarities to both the Hoppers and the humans, peace is made.
On page 138 Jeff finds his brother Ben. He and the rest of his crew had put themselves into deep sleep in the hope of later rescue.
He found the broken needle stuck through the sleeve of the jet suit into Ben's frozen arm.
The fact that the needle broke off in Ben's case puts it in doubt that he will recover, but he does.
This might be Robert A. Heinlein's 1952 juvenile SF novel The Rolling Stones. (It was written ten years before the rock band of the same name was formed.) The UK title was Space Family Stone. The novel has the main character named Stone - in fact all the main characters are named Stone, being from the same family. The story starts on the Moon but most of it is set in a space ship, travelling within our solar system rather than making an interstellar journey. The "spiders" you remember could be the little alien creatures called "Flat Cats" which are cute but breed too fast. The tribbles in the famous Star Trek episode The Trouble with Tribbles were inspired by Flat Cats.
I don't remember a part about a broken needle in someone's arm, but it is many years since I read the book.