Arroceria El Puente for rice dishes, but also fish and a nice whole shoulder of lamb between two. Be careful when booking, they have a bar/cafe just across the street also called El Puente and the two get confused on The Fork sometimes. Best is to walk in to reserve.
30 metres up the street from El Puente is Restaurante Millor, a modest place that only does lunch. €13 for three courses including a drink. You will probably be the only tourists in there. Go at 12:30pm or 3pm if you want to avoid the possibility of having to wait.
Across the road and down a bit is El Txoko de Martin, a semi-retirement project of one of Spain's top celebrity chefs. It's not cheap but it's not crazy prices either, and the food is pretty good without being especially fancy.
Bar Rossellon, not far from the Olivar market, also has a daily menu at a similar price, but with a big bottle of wine and lemonade on the table so you can have as much tinto de verano as you want (if you run out they will bring you more). As with Restaurante Millor, the interior is unprepossessing (most places with "smart" decor are aimed at tourists). The owner looks quite a bit like the late American actor, Dean Stockwell.
The Olivar market itself has several stands where you can have a glass of cava and a couple of tapas at 2pm, or indeed at 10:30am.
There is a small local chain of coffee/pastry shops called Can Joan de s'Aigo that has been running since 1700 (not at their current locations though).
Marc Fosh (fine dining, normally €€€€€) has a relatively affordable lunch menu (only €€€), if you go easy on the drinks. The food is exquisite.