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Beets will naturally grow with some of the root above the soil. If the entire root is trying to form this way, it could mean the soil is too warm (this is a cool season crop) or there may be a lack of phosphorus or potassium in the soil (which helps with root growth). Here is more information on growing beets: https://www.gardeningknowhow.com/edible/vegetables/beets/growing-beets.htm
https://www.gardeningknowhow.com/edible/vegetables/beets/beets-small-or-deformed.htm

Beet root crops need somewhat cooler temps and loose soil to properly form a bulb. If you are noticing more top growth and little to no bulb formation, it is likely due to a fertilizing/nutrient issue. While a soil test would allow you to know for certain, it may be that you have too much nitrogen and too little phosphorus. Lots of nitrogen will result in plenty of leafy growth but if your soil is lacking phosphorus, there will be poor root growth. I would add some more phosphorus, like bone meal, to the soil and give it some more time. Also, make sure the plants are adequately mulched to keep the soil cooler. Here is more information: https://www.gardeningknowhow.com/edible/vegetables/beets/beets-small-or-deformed.htm

There could be a number on insects, rodents, rabbits and even slugs can devour your seedlings.
Here are some links with more information.

https://www.gardeningknowhow.com/plant-problems/pests/insects/kill-garden-slugs.htm
https://www.gardeningknowhow.com/plant-problems/pests/animals/keep-rabbits-out-of-gardens.htm
https://www.gardeningknowhow.com/edible/vegetables/vgen/row-covers-for-garden-plants.htm

Most varieties are ready to harvest 60 days from planting.
When to harvest depends many times on personal preference, some people like smaller Beets.
Look for the shoulder of the Beets protrude from the soil line. If the roots are being pushed up further, they may have been more crowded in planting, and I would go ahead and harvest. If much of the Beet is exposed and the weather is generally quite warm this time of year, it can effect your plants.
For future planting, you can thin your plants when the roots are about 1 inch in size. Then the plants to 4 inches between then. This allows your Beets to grow to a nice full size.

https://www.gardeningknowhow.com/edible/vegetables/beets/harvesting-beets.htm

If more than 1/3 of the tops are missing I would harvest the beets.
The good news is that small beets are delicious!

With some beet varieties, the beetroot part will normally push above ground when they are nearly ready to be harvested.

If that's not normal for your variety, is it possible the soil could have become compacted so that the roots couldn't push into it well? This could also happen with very sandy soil or if the soil temperature was too high. Check whether the soil or any of the conditions changed since last time you grew beets.

Also, check whether any of these problems could be happening:
https://www.gardeningknowhow.com/edible/vegetables/beets/beets-small-or-deformed.htm