To check if Minecraft Realms is down or the server status, you will need to look up the server status through a number of ways on the web.
The first method is to visit a downdector site or service such as downdector.com and look for Minecraft.
At Downdector, you will be able to see if there has been a spike of down reports by users and comments at the bottom of the page of people reporting issues. If you see either of these at the site, chances are the servers are down.
Another way you can check the server status of either Minecraft or Minecraft Realms is to take a peek at any social media for the game. This also means Mojang as they make the game and are most likely going to comment via social media if any issues arise.
That said, check Minecraft and Mojang support socials as an extra step to see if the servers are down. But if you don't see anything of the sort on either downdector.com or social media for the game, it might be an issue client side.
If you think this might be the case, some things you can try is restarting the game, resetting your router, or restarting your pc. You can also test your internet speed by visiting a speed test site like fast.com, and if the results come back saying it's slow, try contacting your ISP.
In most cases, the best thing to do is wait, as it's most likely an issue with the servers, so you will need to wait for the developers to solve the problems.
Check out some of our other awesome Minecraft guides!