The story you’re remembering is Wilson Tucker’s short novel / long novella
“The Year of the Quiet Sun” (first published in 1970; Hugo- and Nebula-award nominee).

How it matches the details you gave

1. How the protagonist gets into the future
   • Instead of cryogenic sleep, Army Captain Addison “Add” Douglass (an African-American Vietnam-era soldier) is one of a small U.S. government team that tests a brand-new time machine in 1978.
   • After a couple of short proving jumps he is sent about twenty years forward, to the year 2000. From Douglass’s point of view it is an almost instantaneous “sleep” and awakening, exactly the way you remembered it.

2. Abandoned underground facility / gathering supplies
   • The time-gate is in a sub-basement under the University of Chicago. When Douglass materialises in 2000 the building is wrecked and deserted.
   • He salvages rations, water, weapons and medical supplies that the project planners had cached for returning travellers.

3. A devastated city that looks like a war-zone
   • Chicago is in ruins: burned-out cars, bullet-scarred buildings, smashed storefronts. The United States has broken up after years of food shortages, riots and guerrilla fighting.

4. Meeting a frightened child, then the child’s family
   • Douglass finds a white boy (David) trapped under rubble, frees him, gives him water and food and gains his trust.
   • David takes him to his widowed mother, Judith, and her two younger children, who are at first terrified because a Black man has suddenly turned up.

5. “It wasn’t a foreign invasion – it was internal”
   • Judith explains that what destroyed the country was not an outside enemy but a bitterly escalating series of internal conflicts—racial violence, food riots, martial law—until civil society collapsed.

6. “What do the enemy look like?” – “They look just like you.”
   • Judith warns Douglass that a band of marauders is coming back. When he asks how he will recognise them, she says, “They look like you.” The attackers are Black militants; Douglass, being Black himself, finally understands why the white family were so afraid of him.

7. Length, publications and era
   • The story is only about 45 000 words (roughly 150–180 paperback pages) so it often appears in anthologies and “best novella” collections from the 1970s onward.
   • First U.S. paperback printing was in 1970 (Ace Books).

Although the mechanism is time travel rather than literal cryogenic sleep, every plot element you recalled—the lone Black protagonist suddenly 20 years out of his own time, the looted underground installation, the ruined city, rescuing a boy, the fearful white family, and the climactic realisation that the “enemy” is people who “look just like you”—comes straight from Wilson Tucker’s “The Year of the Quiet Sun.”