What is the best way to learn Arabic without going to DLI?

The BEST way? Grow up in the middle east.

Start off with Duolingo. It's on the desktop and can be an app on your phone. Going through their basic course will get you the alphabet, some pronunciation and some 1500-2000 words for an initial vocabulary. Its easy, free and has a game like interface with points, and ladders, goals and rewards that reinforce learning when you have a few minutes here and there throughout your day. But for real learning, I would echo what another poster has mentioned: immersion is the best way. It's just difficult to do unless you can replicate the environment 24/7.

Once you have gotten through about 1/3 to 1/2 of the Duolingo content move on to Youtube, to supplement.

Once you have finished about 3/4's, get a well-reviewed grammar text and workbook, commit to spend time in it every day.

Emphasize every avenue: write, listen, type, speak your target language.

Make it your one and only hobby and commit to spending 1-2 hours per day.

Once you have conquered the Duolingo course, google for "entry level modern standard arabic articles" and several helpful resources will pop up, including the Middle East's standard online new resource, but in a beginner's format. I would include a link but when I did so, RallyPoint allowed an ad in Arabic that I can't even read, so I took it out. You will have to find your own link!

All the above will take you awhile. Work hard.

-Semper Fi

The best part about DLI is the immersion.
Try and immerse yourself in the language and culture.
Watch movies, listen to music, take civilian classes, buy an app or program. Etc.