There are many sizes between 5 1/2 and 5. We use 5 full,
and 5 1/2 scant in addition to 5 1/4.
You can size an eternity ring smaller by adding a little material inside the ring to decrease the size a tad without cutting any material out. This works with diamonds eternity rings and does not create a damage issue. It is not beautiful, but no one sees the inside of a ring when it is being worn. It is only a very small amount of metal to alter the ring size by 1/4.
Making an eternity ring larger can be the addition of a diamond or a blank element worn on the bottom of the finger. You can sometimes mask the added metal, but an eagle eye can usually spot it. On a few rings when a small amount of size up is needed, a jewelry will ream out a bit of material and make the ring larger. Again, not elegant, but a practical solution that is invisible when the ring is being worn. It does thin the ring a little which is not desireable. I rarely would recommend that solution and only on an old ring, not a new one.
For enough $$ one can accomplish nearly the impossible when it comes to ring sizing. It is more a matter of what it is worth to YOU to accomplish the alteration. I have seen people with a budget problem wear a band-aid on a ring for years rather than size it properly.