If you know where to look, it can be quite cheap to eat in Paris.  For lunch - stop at any of the bakeries you pass.  Grab a ham and cheese sandwich, or a quiche with a bottle of water and eat on a park bench.  Same thing for breakfast, just stop into any of the bakeries, or even one of the grocey stores and pick up something quick and easy. 

The Marais has quite a few street vendors where you can pick up a quick and cheap lunch or snack.

Grocery Stores and Ready-To-Eat Food

Grocery stores are a good way of feeding yourself.  Making your own sandwiches is the cheapest way of eating.  The boulangeries (bakeries) are the best places to get bread.  There are cheese shops, then meats shops, to fill in the rest.

Outdoor markets, selling fruits, veg, meats, cheese, etc are readily available all over the city.  They seem to set up one or two days per week, so the trick is to find out what day / location the market will be present. A neighborhood market schedule can be found at http://noworriesparis.com/2014/06/19/...

As for cheap ready-to-eat stores ( i.e. if you are familiar with London, then Paris used to be quite a shock in terms of the absence of similar grocery stores with ready-to-eat food), but in the last three years thing have changed.

You'll find dailies called Monop' (small version of the Monoprix supermarkets) where you can buy sandwiches, drinks, salads, etc... and even eat on site. Also some "healthy fast foods such as Bert's and Cojean can, be found almost everywhere now. Same concept as a McDonalds, but with healthy salads, sandwiches, soups rather than fat hamburgers.

WHY NOT PICNIC ? 

If the weather is nice, why not do a picnic? Paris offers a lot of nice places where to do so: the Champ de Mars near the Eiffel Tower, the Jardin des Tuileries, the quais along the Seine, the Parc de Bercy, the Jardin André Citroën and of course the Bois de Boulogne or the Bois de Vincennes.

You can buy cheap food at any food stores ( Monoprix, Franprix, Huit à Huit, Shoppy, G20), and there you will also find bags that keep your food cool (those ones used for frozen food).

Crepe stands are everywhere and provide nourishment for the health nut (fill it with egg and cheese) or can satisfy the sweet tooth (nutella, jam and butter). Some are made with buckwheat and can take the place of lunch. They range in price from 2 euros to 5.

If you want to make it typically French, buy a saucisson (dry sausage) some paté or rillettes, some good cheese, cherry tomatoes and fruits and buy one or two fresh French baguettes. And do not forget a good bottle of wine, the plastic glasses to go with and of course a bottle opener!