How much barley are you hoping for? I'm not familiar with all grain brewing, but a quick search seems to suggest that 5 gallons takes 5-10kg of barley (malted and not)?
Under normal growing conditions, ie the average farm yield, that would give you 2.84kg of barley. Normal growing conditions is not by fences, not in a lawn, and is definitely not with mowing. The lawn isn't only competing for sunlight, which it is, but also but water and nutrients. It also helps create a humid environment favorable to harmful (for the plant, but also for you) fungal diseases. I don't mean to be a party pooper, and with most growing projects the worst that can happen is typically "oh well, no yield, I'll have to buy some instead", but cereals like barley have an uncommon danger: DONs, also known as vomitoxins, which are mycotoxins caused by fusarium wilt that can be a serious health hazard. It's part of why one should not take cattle feed barley to make beer, because the DON limit is higher for cattle feed than it is for human consumption. And growing barley in a lawn, by fences, seems like the ideal conditions for fusarium to boom.