Half-bloods could be Death Eaters, they weren’t only pure-bloods.
There’s nothing indicating that half-bloods aren’t allowed to join the Death Eaters - they only really hate Mudbloods. When they take over the Ministry, they lock up Mudbloods, but anyone who can prove they have a close wizarding relative is exonerated. Though she’d of course be somewhat biased since she’s Mudblood and strongly against their cause, so she’d be glad to think them all liars, Hermione also suspects that most of the Death Eaters are half-bloods, since there aren’t enough pure-bloods left for the Death Eaters to all be pure-bloods. As pure-bloods are indeed very rare, it seems likely that at least some of the Death Eaters would be half-bloods.
“The Death Eaters can’t all be pure-blood, there aren’t enough pure-blood wizards left,’ said Hermione stubbornly. ‘I expect most of them are half-bloods pretending to be pure. It’s only Muggle-borns they hate, they’d be quite happy to let you and Ron join up.”
- Harry Potter and the Half-Blood Prince, Chapter 12 (Silver and Opals)
Also, J.K. Rowling said in an interview that Snape having been a Death Eater clearly showed that he wasn’t Muggle-born, as they’re only in very rare circumstances allowed to join the Death Eaters. This statement implies that his blood status was one that wouldn’t be an obstacle to him joining.
Apart from Harry, Snape is my favourite character because he is so complex and I just love him. Can he see the Thestrals, and if so, why? Also, is he a pure blood wizard?
Snape’s ancestry is hinted at. He was a Death Eater, so clearly he is no Muggle born, because Muggle borns are not allowed to be Death Eaters, except in rare circumstances. You have some information about his ancestry there.
- Edinburgh Book Festival (August 15, 2004)
Therefore, it seems likely that half-bloods could be Death Eaters, and no special exceptions were made for Snape. That statement also implies that there are circumstances, though rare, that would cause a Muggle-born to be allowed to be a Death Eater. However, from Rowling’s statement about Snape’s blood status, it implies he was of a status considered acceptable by the Death Eaters.