Big companies can still buy little companies to take on their lawsuits, or they can provide funding & resources to fight a legal battle as part of an investment.
I suspect that Google bought YouTube in part to avoid precisely this situation. If Viacom had won their lawsuit against YouTube, it would have endangered Google's core business: they would become liable not just for copyrighted material that they know about, but any copyrighted material on their servers, which encompasses the whole web. So they pay $1.6B for a tiny startup and a big lawsuit, and then use their full legal resources to fight (and eventually win) the lawsuit.
(There is possibly an interesting get-rich-quick hack in there: do something in a legal grey area that lots of big companies are doing too, as part of their core business, and then get sued. It is then in the big company's interest to acquire you and fight your legal battles, lest they get put out of business by an unfavorable precedent. You're playing with fire in this case, though, since if the big company doesn't come to your aid you have a multi-hundred-billion dollar lawsuit hanging over you.)