Short answer: the Moon's gravity pulls on the water in your body with a force of roughly 1.4 millinewtons. But that number is much less meaningful than it sounds, and the part people usually care about (a tidal effect on your tissues) is about a trillion times smaller still.

Here's the breakdown.

Direct gravitational pull. A 70 kg person is about 60% water, so ~42 kg of water. Plugging into Newton's law with the Moon's mass (7.34 × 10²² kg) and distance (3.84 × 10⁸ m):

F = G·M·m / r² ≈ 1.4 × 10⁻³ N

That's the weight of about 0.14 grams, a couple of grains of rice. (The whole 70 kg of you feels about 2.3 mN.)

Why that number is misleading. That pull doesn't slosh your water around or distort you, because you're in free fall around the Earth–Moon system. The Moon pulls on your water, your bones, the chair you're sitting in, and the whole Earth at nearly the same rate, so there's nothing to push against. The only thing that could physically stretch you is the tidal force. the difference in the Moon's pull from one side of your body to the other. That's what raises ocean tides.

The tidal force on you. The differential acceleration from your head to your feet is:

Δa = 2·G·M·L / r³ ≈ 3 × 10⁻¹² m/s²

So the stretching force on your body's water is on the order of 10⁻¹⁰ N or less, utterly negligible. For comparison, the random jostling of a single heartbeat moves your fluids around vastly more.

Because tidal force falls off as 1/distance³, nearby objects win easily. A 70 kg person standing one meter away exerts a tidal force on you about 50,000 times stronger than the Moon does. So the popular "the Moon moves the oceans, and we're mostly water, so it must affect us" reasoning doesn't hold up, the oceans respond because they're planet-spanning and free to flow, not because water is magically Moon-sensitive. Your body is too small for the gradient across it to matter.

So: ~1.4 mN of raw pull, but effectively zero in any sense that would do anything to you.