Bleak Falls Barrow

Solution 1:

This quest is the third step of the skyrim Main Quest. You will not be able to progress in the main quest without finishing it.

Dragons only start to appear after after the fourth step, Dragon Rising. This mean that you will not be able to use any dragon shout without finishing Bleak Falls Barrow, as you need to obtain dragons souls (thanks Mixxiphoid for the correction)

Solution 2:

The Dragonstone is the third chapter of the Alduin questline, but it is also part of a "beginner dungeon". The game flow intends for you to, most likely, end up in Riverwood before you go anywhere else. The general store there is run by a brother (Lucan) and sister (Camilla) who will send you to recover a missing item. That missing item happens to be required to reach the Dragonstone; a golden claw. Lucan Valerius issues the quest, but it is also generated when you first enter the Barrow. The multiple pronged hole there is intended for you to insert the claw that you were sent to pick up for the side quest. All such gates require one of these claws, and the palm of the claw has the puzzle solution (check it in your inventory by viewing the object) for its respective gate. The claw itself is in the inventory of Arvel the Swift, a thief you cut down from some spider webbing and (unsurprisingly) gets himself killed when he tries to flee without giving you the claw. Just take it off of his corpse.

You can accidentally lock yourself out of the Dragonstone room at the end of the Hall of Stories (the name for the hallways leading to the ancient Nord puzzle gates, note the glyphs on the walls). If you turn in the quest to Camilla and her brother, the claw is removed from your inventory and placed on the counter in plain view. From here, the only way to get in is to steal the claw back by sneaking in during the night or otherwise distracting the family. Please note that if you do this, even if you later marry into the family, Camilla will pay for mercenaries to attempt to take the claw back from you. Inversely, if you marry into the family and THEN want to take the claw, the claw is marital property and you can take it without it being flagged stolen and without Camilla hiring thugs to come after you.

This is one of two claws I'm aware of where you can lock yourself out. The other belongs to a woman in Winterhold who can die if she is attacked the first time you enter town. She sells the claw after a dialog option is pursued, so make sure to pick it up the first time you are in town.

I can't stress enough that you should never sell a door claw. Some mods will re-use these claws, and I think I recall one situation where more than one claw is used from other dungeons. Money will rapidly become easy to obtain in Skyrim, so don't let the high value of the items fool you into thinking you're getting easy money; later you'll be selling items you've enchanted for so much money that you'll buy the store out of its other stock first just so the merchant can afford what you are selling.

If you are on PC, the console code to recover the claw to your inventory is

player.additem 39647 1

Using console commands to obtain quest items can play wonky with the quest, so use it only if you lost or sold the claw and can't buy it back. If you stole the claw and got arrested after, it's in the evidence chest of the associated hold.

If your issue isn't related to the item, and is instead related to a bug, don't just use the console to fix it. This quest is a hingepoint for the main quest and a side quest, and you can really screw up the main quest scripts and sequence by using quest advancement or quest item commands to skip sections of it. I strongly recommend just reloading a prior save if the problem is bug related, and adding what the bug is to your question here.