Putting aside that he's using puts (which caps the downside), with a classical short sell you don't need the current value of the stock liquid, but typically some percentage. That represents how much the stock can go up before you get the margin call. A short position is fundamentally unlike a long one, in that the future price movement of the underlying is bounded below at $0 but not bounded above.

The sources I can readily find put Burry's current net worth in the neighbourhood of $300 million. Depending on the regulations, he probably actually could put his entire life savings into a short of this magnitude. Of course, that's sort of like having your entire 401(k) in a -4x levered ETF, even worse because it's on individual stocks.

The filing doesn't appear to disclose strike prices or expiration dates. But my guess is that he loaded up on very cheap puts (low strike price) to hedge against the apocalypse (low probability of winning big to cover losses from everything else; high probability of just paying some insurance money). The same form shows bullish positions in other sectors — health care, finance and energy, as well as some corporate bonds. Given what the portfolio in the filing looks like overall, it's hard for me to imagine him being willing to risk more than a few million on this.

Read the article. He used options, which can't be margin called because they're already paid in full.

That’s why he did options instead of directly shorting the stock. You’re betting against them but with a capped downside effectively.