Can I sell a LEGO set without the box
Yes. You can sell a LEGO set without the original box. Say so in the listing, price it as used rather than sealed, and include the instructions and minifigures if you have them.
Yes. You can sell a LEGO set without the original box. Buyers purchase used sets without boxes every day, as long as the parts are there and the listing says the box is gone. Do not describe it as sealed, and do not use a stock photo of a mint box if yours was thrown away. Price it against other used sales of that set number, not against factory-sealed sales.
On Brick Deals, list it as Used - No missing parts if every part and minifigure is present, or Used - Missing some parts if anything is gone. The box is separate from completeness of the build. Mention the missing box in the description in the first lines, not at the end. Start selling and add photos of the actual bricks, the instructions if you have them, and the minifigures.
How much the box is worth
For a set that is still in the shops, a missing box matters less, because buyers are often happy to build it. For an older or retired set, collectors pay more when the box is present and clean, and a crushed box already reduces a sealed sale. A used set without a box should cost less than the same used set with a tidy box and the instructions. It should not be given away. Check the range on the homepage with the box marked as opened, then take a further step down if you have no box at all, and compare with catalogue listings that say no box.
Keep the box if you still have it and it is not filthy or falling apart. It makes the set easier to identify, it photographs well, and some buyers want it even if they will never display it. If the box is badly damaged, say so. A torn box is still useful information. It is not the same as no box. The delivery box you use to post the set is a different thing: you still need a sturdy carton to post the bricks, even when the original LEGO box is missing. See how to pack and post a set.
What to include instead
Instructions matter more once the box is gone. They prove the set number and let the buyer rebuild it. If the booklet is missing too, say that clearly. Many sets have digital instructions from LEGO, but do not assume the buyer wants to hunt for them, and do not assume every old set is available that way. Minifigures, sticker sheets, and any printed parts should be photographed loose so the buyer can see they are included.
Sort the bricks before you photograph them. A picture of a full set of bags, or of the model built, answers the question the box would have answered. If you are not sure the set is complete, do not claim that it is. List it as missing some parts and describe what you have checked. That is fairer, and it avoids a dispute after delivery. Buyer Protection covers sets that are not as described.
When to keep the set together
A set without a box is still a set if the parts belong together. Splitting it into loose bricks usually makes it harder to sell, not easier, unless the minifigures are the only valuable part. Name the set and the set number in the title so someone searching for that model can find it. Without the box, the title and the first photo do all of the work.
Look at how to price a used set before you publish. You keep the item price you choose. The buyer pays postage and the 5% protection fee, as set out on the fees page. A no-box set priced like a sealed one will sit. A no-box set priced like other used copies, with clear photos, is a normal sale.