Pac-Man is a Pac-Person, a presumably biological species that lives on Pac-World (Or PacLand, as it's called in the 1982 Pac-Man TV Series):
Likely, PacLand is a region or continent of Pac-World
His modern depiction has been around since Ms. Pac-Man arcade game and the original Pac-Man TV series, although it varies slightly from the cabinet art on the original arcade machine.
Left: 1980 Pac-Man arcade cabinet featuring a Pac-Man with legs, but no arms. Right: Pac-Man as featured in the latest Smash Bros. game, matching up with the more fully-featured and limbed version seen since 1982
Since 1982, when Ms. Pac-Man was originally released, we've known there are additional Pac-People, and the idea that they have arms, noses, and eyebrows came into play.
Original 1982 Ms. Pac-Man cabinet, showing the general design for yellow Pac-People that became here to stay, next to a closeup of related game art.
we've since also seen other family, such as additional children and uncles. (However, the newer TV show is set when Pac-Man was a teenager, so before his time with Ms. Pac-Man).
Some of his kind are known to grow hair:
Picture of mustachioed Professor Pac-Man from 1999's Pac-Man World, a 3D adventure game for PlayStation 1
And although they appear biological, how they're "born" is unclear. The main evidence we have is that storks can deliver infants, but we don't know exactly where the infants come from:
The cutscene from the original Ms. Pac-Man game showing a stork delivering Jr. Pac-Man
Another indicator that they're biological is that their big mouth and tongue are also used to eat more than just power pellets (or bonus fruit), as his species seems to be able to consume a wide variety of foods (and for pleasure, not just energy):
"I really should work on my self-control" - Pac-Man, immediately prior to eating a whole hamburger in one bite, followed by pretty much everything else on the table.
There are also Pac-People in a variety of colors and shapes, as seen in PAC-MAN: The Ghostly Adventures.
A note is that in the original TV series, all the Pacs we see are some sort of yellow color, even those in the final episode "Happy Pacs-Giving" (where Miles Pac-Man named Pac-Land). In the new series, yellow Pacs are considered rare. Additionally, as we see in that same Pacs-Giving episode, Pacs consume a variety of foods (as evidenced by their table full of a traditional "Pacs-Giving" dinner).