How can one have sex with a woman without being accused of rape? Is a written contract enough?

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A lot of people don't seem to understand how trust is supposed to work. You don't have sex with someone outside of an atmosphere of mutual trust. False rape accusations are the least of the risks you're taking on if you do that. Other things women can do to you:

Poke holes in your condom.
Falsely claim she's on the pill.
Falsely claim she's drug and disease free.
Misrepresent her personality and destroy your stuff and life when you do something she doesn't like.

The list of things you could do to her if she has sex with you without trusting you is even worse.

Don't have sex with someone you don't know well enough to be able to trust.