- Are there techniques, regardless of practically how difficult they are, of possibly measuring domestic work?
Yes, measuring household gross production (GDP measures gross product not amount of work), is just matter of estimating how much households produce, in a given unit of time (e.g. quarter, year... etc), and what would be value of that output on the market (or with income approach how much would household get if the household production was supplied to the market). Both can be estimated via various econometric techniques.
Nowadays econometrics is too broad to provide overview of all possible techniques that could be used. However, a commonality among those techniques is that they estimate the value of household production by inferring it from value of observable output. For example, one simple method just looks at value of production that is done in market (e.g. nanny services) and then assigns the same value to the amount of that kind of output produced by household in house. Other approaches try to measure the opportunity cost of households for home production and so on. You can have look at Dulaney et al. (1992) for overview of various methods, but there is quite a lot of heterogeneity in approaches that can be applied here especially when it comes to the details.
- Have such estimates been made by researchers, maybe at scales smaller than the national economy?
Well this was already done on scale of national economy. In fact, Bureau of Economic Analysis (BEA) already provides estimate of value of US household production that is estimated in a way to make it complementary to GDP (i.e. you could add it on top of it - although that does not mean that you should as it is also less precise than just GDP itself, so it depends on a particular research if that is worth while sacrifice). In fact BEA even produces household production adjusted GDP (see their article on that here). Although, most statistical offices around the world don't provide such estimates yet.
Beyond statistical offices, individual researchers were providing estimates for value of household output already back in 70s and possibly even earlier (e.g see Weinrobe, 1974 as an early example), with various degrees of precision and success. However, I don't know any study that would do that on a regional level (and quick google scholar search did not revealed any either), but as mentioned at the beginning that would be actually much harder to estimate.
- What are some theoretical approaches to understanding this? I remember Amartya Sen developed a model about intra-household trade. Is this related? Are there other such well known models?
This seems unrelated to the above questions. You can estimate household output without having a theoretical understanding what determines how much output will be produced by households. This being said, in economics there is a specific subfield of economics of family. However, it is beyond scope of any reasonable SE answer to provide literature review of whole field, although generally speaking the theories of household productions are somewhat analogous to the regular ones (e.g. households will have production functions, their stocks of labor and capital, costs, home production competes with production that could be gotten from market, they will try to optimally allocate their resources to maximize utility etc.). You can have look at overview of various theories of household production in Browning & Martin Economics of the Family or Menchik Household and Family Economics or the classic text Gary Becker Family Economics (although slightly out of date now).