Press your lips together, add vocal cord vibration, then release.

Americans pronounce barrel as BA-ruhl (/ˈbærəl/). The L in "barrel" is a dark L — the back of the tongue rises toward the soft palate, adding a small "uh" quality before the L. This is called the Dark L vs Light L, a hallmark of natural-sounding American speech. It comes out as BA·ruhl. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "He uses a rain barrel to collect water for his garden".
Record yourself saying "barrel" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 5 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Curl or bunch your tongue without letting the tip touch the roof of your mouth. Brace the sides of your tongue against your upper back teeth, and round your lips slightly.
Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
Keep the tongue tip down and pull the back of the tongue up toward the throat. The 'dark' sound comes from the back.

The textbook way isn't wrong — it's just not how anyone actually says it.
The L in "barrel" is a dark L — the back of the tongue rises toward the soft palate, adding a small "uh" quality before the L. Dark L adds a small schwa-like "uh" before the L. The back of the tongue lifts toward the soft palate.
Stress falls on the first syllable, not the others. Stretch BA — keep everything else short and quick.
Don't pronounce the first syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.