A lurcher can be quite a bit smaller than a greyhound and they make marvellous pets. They often end up in rescue centres.
As for getting a cat as some have suggested, from personal experience they are much more trouble than a dog

When it's time to go to the vet, with a dog you just pick up their lead, it's "Oh goody, we are going out" and off you go
With a cat, you have to prepare the carrier beforehand, so the cat doesn't see it, then you have to creep up behind them, preferably when they are asleep (which is most of the time), and preferably wearing gloves then grab them before they can run and squash them kicking and screeching into the carrier. If you're lucky, you'll get the door shut before they wriggle free and race off to hide goodness knows where for the rest of the day.
(I love her really

)