Is it too late for me to learn an instrument?
No. It's never to old to learn anything. Having the determination and persistence to continue is the hard part. It might be harder for you to learn as your brain isn't as flexible as it was. Music is a language and it will take work to learn. Don't get discouraged though!
Can I learn it by myself? Hiring a teacher is out of the question. I plan to learn by myself. If this is not possible should I give up?
Of course you can. There is a source right here for you to ask questions regarding technique, etc. But, I would recommend getting a basic theory book and an instrument book (make sure it isn't band oriented). These will give you a path to follow, then supplement with videos on YouTube, etc for other things you want to learn.
There are some downsides to self learning especially if you pick violin. You will probably end up with bad posture and other problemspossibly end up not even hitting a correct note. You can probably find some sources on the internet on ways to prevent and on what proper posture and finger placement is. And, etca beginner with a violin sounds VERY bad (You've been warned. :)).
My choices are violin and piano. Which is easier to learn?
My personal opinion is that the piano is easier to learn. Violin requires precise hand movements (not that the piano doesn't, but less so). It will also be easier to play, note- wise, because it should already be in tune. I also think the piano is more versatile, you rarely hear a pop song with a violin, but every song has a chord progression that can be played on the piano as you or someone else sings along.