There's already plenty of legally questionable stuff in Bitcoin's blockchain:

https://news.ycombinator.com/item?id=16617136

Thus, it would be better (if it could be afforded --- see some of the other comments on cost estimates) to continue there, or perhaps one of the other altcoins that's already in use for monetary purposes.

It's very incidental, though, and most of it could be purged with mild effort.

The whole point of a blockchain is that it's append-only. Each block has a hash of the previous block. If you wanted to modify a Bitcoin block from a year ago, you'd have to do the same computation as a year worth of all miners.

The blocks are hash trees, and you don't need to store the whole tree. Purging depends on that, not altering the block history.

What if all servers would hold some random bits and no one would have any whole file (only the clients would be getting whole files)

The talk of holding "random bits" reminds me of the idea behind the Owner-Free File System:

https://en.wikipedia.org/wiki/OFFSystem

Basically people would be hosting random-looking blocks of data, but users could be instructed to XOR two carefully chosen blocks together, producing a meaningful file.

Because of the nature of XOR, if Alice sees that Bob is hosting a file F, she could generate and host a file G, such that F XOR G produces a banned file, providing plausible deniability for Bob (as long as he can point to a file H such that F XOR H produces an innocent file).