I read an article about this a while ago. Apparently the main obstacle is the formation of destructive ice crystals. They can avoid this by cooling fast enough to extreme low temperatures, which is already used in practice on very small things:

This process is now routine for eggs, sperm, and embryos, which are cooled to −196 °C in less than two seconds and can be used even after decades in storage.

There have been recent breakthroughs in preserving whole organs, albeit not indefinitely so far:

In new research, one team has been able to supercool the kidneys of pigs—animals whose organs are of a similar size to human ones—and preserve them for days. The kidneys survived being stored at −4 °C (25 °F) and eventually reimplanted back into pigs.

There’s also techniques that don’t rely on cooling:

Another way to prolong the lifespan of an organ is to use a machine that perfuses it with nutrients, mimicking what happens inside the body. Machine perfusion devices have become more commonly used over the last decade or so and are typically used to maintain livers and kidneys for up to about 24 hours.

They’re not close to bringing frozen people back from the dead though:

When Coles’s friend Greg Fahy, a cryobiologist, studied pieces of his brain years later, he found that the brain cells, which had shrunk, “bounced back” once they were rewarmed. But that doesn’t mean the cells are alive, or that it might one day be possible to reanimate the brain. As Matthew Powell Palm of Texas A&M [the guy who did the pig organ thing] told me at the time: “There are so many ways those neurons could be toast.”