If something is countable (e.g. disk space) but is limited by some quota. What you will ask: "how much" or "how many" quota do I have?"

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    Neither. You would need "How big/large a quota…" or just "What quota (do I have)?" – Robbie Goodwin May 13 '18 at 21:41

When you can count something, use many:

How many people were at the party?

When you can't count something, use much:

There was too much paint on the walls.

A quota is a limit; it is not countable. What it limits might or might not be countable. So with quota, you should use much:

How much is my quota?

However, it's also very common to say:

What is my quota?

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