Lucy gleefully tells Ethel that she never mentioned transportation to Ricky and that if comes home on any public transportation bus, subway, etc. they win the bet. However, Ricky could have used Manhattan's Third Avenue Elevated line get home. The El, which ran the length of Manhattan's Eastside, was still in service in 1952. The line was around in 1900 which was also the year it was electrified. It was torn down in 1956.
Ricky says that he and Lucy will compromise on the terms of the bet so she can use the stove, however electric stoves were invented in 1896, so she would have been able to use the stove under the original terms of the bet anyway.
When Lucy is kneading the dough for the bread she gives the impression that it gets heavier after each rest period. The difference in the size of the dough is because of the carbon dioxide produced by the yeast, the actual weight would be no more than that of the original ingredients.
After Lucy and Ethel put the bread dough in the oven to bake, Lucy suggests they kill time by playing Canasta. Canasta was not invented until the late 1940's and thus they couldn't be playing it if they were following the rules to not live with anything that was invented post 1900 (though it's possible that the rule only applied to household items and "modern conveniences".)
When the inspection committee is viewing the Ricardo's apartment, the electric lamp in the background is on. Since the couples had agreed to live like pioneers the lamp should not have been on. Lucy had even went to the trouble of buying lanterns to avoid using electric lighting.
When the bread is coming out of the oven, there is a space behind the oven where you can see the bread being fed through a hole in the wall.
When the bread comes out of the oven, the counter, which Lucy has been using to knead the bread, is gone.
When the investigating committee enters Lucy's kitchen to inspect it, the two women can be seen standing just inside the swinging door waiting for their cue to re-enter the living room.
When the investigating committee goes into Lucy's kitchen to look around while the Ricardos and Mertzes wait in the living room, the two women just go through the swinging door and stand right inside it without actually going further inside the kitchen. They are visible when the door swings shut.
The girls actually lose the bet early on. After putting the bread in the oven they go to play canasta, but canasta was invented in 1939 and only introduced in the US in the late '40s.
Lucy and Ethel put on pre-1900 dresses, but they still wear post-1900 makeup (lipstick, eye shadow, false eyelashes).