Yes both beetroots and beets do eventually go bad just like other foods.
Beetroot or beets should last for 3 to 7 days at room temperature.
However it's best to eat the beets or beetroot within a few days and keeping in the refrigerator for the 3 to 7 days can help them last a bit longer.
Or store the beetroot or beets in a cool dark place such as in a cellar if you have one.
But it's best to eat the beetroot or beets within a few days of purchasing them.
If you're growing beets in a garden then only pick the ones you plan to eat and then leave the rest in the ground to make them last.
But remove the beets from the ground before winter.
White beetroot and white beets are OK to eat and they are edible and safe.
Sometimes the beets or beetroots will turn white inside and that is normal and nothing to worry about and the white beetroot or white beets are not an indication of bad beets or bad beetroot.
The taste may be a bit different but white beets and white beetroots are entirely safe to eat and you won't get sick or have any health issues when eating white beetroot or white beets.
Some beets can be white inside while being red outside and that is normal and the beet is still good and safe to eat.
Some beets grow to have a white inside while the outside is red or even sometimes other colors.
But the white beet is okay to eat when it's white inside.
Having a red beet that has a white inside is still a good beet and that is just how some beets grow.
Sometimes the red beets turn white inside because the soil was allowed to get too dry and the weather was too hot.
The white insides of a beet or beets are not an indication of bad beets but it usually means that they may not have gotten enough water.