Prepare your pressure canner, jars, lids, and bands according to the manufacturer's instructions. Keep jars hot until needed.
Wash and halve the celery stalks lengthwise, peel and halve the carrots, and halve the onions root-end intact so they hold their shape. Bundle the peppercorns, bay leaves, and caraway seeds in a piece of cheesecloth tied with kitchen twine to make a spice bag. Set aside.
Combine the paprika, salt, and mustard powder in a small bowl. Toss the meat cubes with the spice mix in batches until evenly coated. Heat some of the oil in a large heavy-bottomed pan over medium-high heat and brown the meat in batches, adding more oil as needed. Transfer each batch to a large pot with the prepared vegetables as it finishes.
Deglaze the frying pan with the water, scraping up all the browned bits from the bottom, and pour that liquid into the pot. Add the spice bag and vinegar. Bring to a boil, then reduce heat and simmer covered for about 1 hour until the meat is just tender but not falling apart. Alternatively, pressure cook on high for 7 to 10 minutes.
Remove the spice bag. Remove and discard the vegetables, or leave them in if following the Hearty Soups Protocol. Ladle the hot goulash into prepared jars, making sure each jar has a good balance of meat and broth. Top with boiling water if needed. Leave 1 inch headspace. Remove air bubbles, wipe jar rims, and apply lids and bands to fingertip tight.
Load jars into the preheated pressure canner. Lock the lid, vent for 10 minutes, then bring to pressure. Process pints for 75 minutes and quarts for 90 minutes, adjusting pressure for altitude. When processing is complete, turn off heat and allow canner to depressurize naturally. Wait a few minutes, then remove jars and cool on a towel-lined counter for 12 to 24 hours. Check seals, label, and store in a cool dark place.