I'm not math-y enough to even begin to go about proving this, but just based on logic I believe you can just cut the circle in half to give you a semicircle with a diameter of $1$ unit, which would have an area of $\tfrac{1}{8} \pi$ (a circle with a _diameter_ of $1$ has an area of ${\pi \cdot\tfrac{1}{2}}^2 = \tfrac{1}{4}\pi$).

This will fit the longest possible layout along the diameter, and I don't think it should be possible for the snake to lie in such a way as to break out of the semicircle.

If anyone can provide any kind of proof (or disproof), feel free to edit my post.

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Here's an illustration of the blanket, provided by @MartinFrank. The red lines indicate possible snake positions.

![enter image description here][1]

[1]: http://i.stack.imgur.com/ddTOe.png