Sounds like the Hogben series by Henry Kuttner and C. L. Moore, which was also the answer to the old question "Hillbilly-type family with supernatural powers". Here is a video ad by Neil Gaiman for The Hogben Chronicles.
You probably read the first story in the series, "Exit the Professor", originally published in Thrilling Wonder Stories, October 1947 which is available at the Internet Archive (click here for download options). Any of these covers look familiar?
They were of varying ages, ranging from relatively young to thousands of years old. They had lived in Europe before immigrating to the American colonies.
Grandpaw's the oldest one of us all and he gets kinda mixed up in his language sometimes. I guess the lingo you learned when you're young sorta sticks with you. One thing, he can cuss better than anybody I've ever heard.
"Shucks," I said, "I was only trying to help."
"Thou puling brat," Grandpaw said. "'Tis thy fault and thy dam's. For building that device, I mean, that slew the Haley tribe. Hadst thou not, this scientist would never have come here."
"He's a perfesser," I said. "Name of Thomas Galbraith."
"I know. I read his thoughts through Little Sam's mind. A dangerous man. I never knew a sage who wasn't. Except perhaps Roger Bacon, and I had to bribe him to—but Roger was an exceptional man. Hearken:
"None of you may go to New York. The moment we leave this haven, the moment we are investigated, we are lost. The pack would tear and rend us. Nor could all thy addlepated flights skyward save thee, Lester—dost thou hear?"
[. . . .]
"How old is your grandfather?"
"Gosh, I dunno."
"Homunculi—mm-m. You mentioned that he was a miner once?"
"No, that was Grandpaw's paw," I said. "Tin mines, they were, in England. Only Grandpaw says it was called Britain then. That was during a sorta magic plague they had then. The people had to get the doctors—droons? Droods?"
"Druids?"
"Uh-huh. The Druids was the doctors then, Grandpaw says. Anyhow, all the miners started dying round Cornwall, so they closed up the mines."
"What sort of plague was it?"
I told him what I remembered from Grandpaw's talk, and the Perfesser got very excited and said something about radioactive emanations, as nearly as I could figger out. It made oncommon bad sense.
"Artificial mutations caused by radioactivity!" he said, getting real pink around the jowls. "Your grandfather was born a mutant! The genes and chromosomes were rearranged into a new pattern. Why, you may all be supermen!"
"Nope," I said. "We're Hogbens. That's all."
I believe the youngest, just called Baby, was the most powerful and least human looking, and lived in the attic. He slept most of the time.
The baby was called Little Sam:
Time we ran off the Haley boys with that shotgun gadget we rigged up—only we never could make out how it worked, somehow—that time, it all started because Rafe Haley come peeking and prying at the shed winder, trying to get a look at Little Sam. Then Rafe went round saying Little Sam had three haids or something.
Can't believe a word them Haley boys say. Three haids! It ain't natcheral, is it? Anyhow, Little Sam's only got two haids, and never had no more since the day he was born.
[. . . .]
"S'pose I go to New York with you, like you want," I said. "Will you leave the family alone?"
He halfway promised, though he didn't want to. But he knuckled under and crossed his heart, on account of I said I'd wake up Little Sam if he didn't. He sure wanted to see Little Sam, but I told him that was no good. Little Sam couldn't go to New York, anyhow. He's got to stay in his tank or he gets awful sick.
