Look around at the desirable plants nearby, and read the label on 2,4-D to see if that will have undesirable effects on any of them. Grasses and some woody plants are immune to 2,4-D if it's used in lawful quantities (you must obey the instructions on the label). Poison ivy, however, is killed good-and-plenty by it.
You'll still need to leave the area alone for several months... the irritants in poison ivy take a long time to degrade. DO NOT BURN IT.
Having had my life changed by herbicide I don't recommend them casually. 2,4-D is one of the oldest, and its toxicity (or rather, its lack) is extremely well-understood. It's patent-expired/generic, it's not tied to a popular brand name the way Roundup is, so no company has any stake in it. Nobody's lobbying or suppressing medical research or burying lawsuits or forcing their way around safety laws that limit the dangerous stuff to licensed applicators. In fact the industry would prefer 2,4-D go away so instead you'll buy products whose patents are still valid. In fact it's often the "weed" part in proprietary "Weed-and-seed" or "3-way" combo products that include fertilizer and/or pre-emergent broadleaf killers.
The deep burgundy concentrate like you get at farm supply is dangerously acidic, and can cause the kind of harm that acids cause. But you'd probably buy the lighter colored pre-mixed stuff at the home store.
Also, 2,4-D is not Agent Orange. The scary ingredient in Agent Orange was 2,4,5-T, totally different chemical. If manufactured stupidly, an impurity was created called dioxin. That's a 2,4,5-T problem, not a 2,4-D problem. Needless to say, you won't see 2,4,5-T at the box store.