As a rule of thumb, if you're not completely technologically illiterate or a trader then you are better off withdrawing your money and storing it offline. Exchanges lose their customers' funds all the time.
In 20 years maybe it won't be so uncommon to have 20TB lying around, but I know I don't have a lot of things on my computer that take up more than 1TB and only ever grow. I certainly won't have two of them...
I'm not being disingenuous, I had coins in a cold offline wallet, and it was a serious endeavor to get them out to where I could spend or cash them. I had to spend days downloading a block chain and the only thing that kept me from freaking out was that I knew this would work, having done it plenty of times when the chain was shorter. I couldn't find any way to retrieve coins from a wallet.dat without downloading the block chain first. I had thousands of dollars tied up for days while I waited.
I can't imagine what it would be like when the block chain was 8x its current size, or if I had more than one cold wallet and I needed to empty all of them in any kind of a hurry (like, in time to capitalize on a bubble before it pops!)