There is more than one type of UUID, so "how safe" depends on which type (which the UUID specifications call "version") you are using.
Version 1 is the time based plus MAC address UUID. The 128-bits contains 48-bits for the network card's MAC address (which is uniquely assigned by the manufacturer) and a 60-bit clock with a resolution of 100 nanoseconds. That clock wraps in 3603 A.D. so these UUIDs are safe at least until then (unless you need more than 10 million new UUIDs per second or someone clones your network card). I say "at least" because the clock starts at 15 October 1582, so you have about 400 years after the clock wraps before there is even a small possibility of duplications.
Version 4 is the random number UUID. There's six fixed bits and the rest of the UUID is 122-bits of randomness. See Wikipedia or other analysis that describe how very unlikely a duplicate is.
Version 3 is uses MD5 and Version 5 uses SHA-1 to create those 122-bits, instead of a random or pseudo-random number generator. So in terms of safety it is like Version 4 being a statistical issue (as long as you make sure what the digest algorithm is processing is always unique).
Version 2 is similar to Version 1, but with a smaller clock so it is going to wrap around much sooner. But since Version 2 UUIDs are for DCE, you shouldn't be using these.
So for all practical problems they are safe. If you are uncomfortable with leaving it up to probabilities (e.g. your are the type of person worried about the earth getting destroyed by a large asteroid in your lifetime), just make sure you use a Version 1 UUID and it is guaranteed to be unique (in your lifetime, unless you plan to live past 3603 A.D.).
So why doesn't everyone simply use Version 1 UUIDs? That is because Version 1 UUIDs reveal the MAC address of the machine it was generated on and they can be predictable -- two things which might have security implications for the application using those UUIDs.