If the block chain is ever growing chain, doesn't it run out of
storage? Can't this chain be removed permanently?
No, the whole point of a blockchain is that everybody has to store a copy of the blockchain. That's what makes it decentralized, so there's no authority that stores the blockchain and "watches" over it.
The current bitcoin blockchain (as of 9th April 2019) is $\approx 211$ GBs large.
Does the validation require all peers to agree that the chain is valid
or only the majority? What if a malicious peer joins the group and
gives false validation that a valid block is invalid? Does this break
the entire chain?
Individuals always agree on the longest blockchain, since that is the one with the most work behind it. If a malicious person wants to create a longer and different blockchain he would have to outcompute everybody else who is working together honestly and that is not very feasible. The malicious person needs to have more than $50\%$ of the total computing power to fool everybody.
It could actually be possible, that a fork happens if two miners create a new block. If that happens then you just wait to see which one of the forks gets more blocks attached to it. This would then be the longest block and you then agree on this fork. The other fork can be discarded.
If you still have some questions, then I would recommend to watch this video from the YouTube channel 3Blue1Brown, it explains it quite well.