There are repeated measurements for each individual.
You say that these measurements are taken along a day. If you are not assuming that the order of the measurements has an importance, then a common way to model your entire dataset consists in attributing an exchangeable multivariate normal to the series of measurements for each individual $j$ of each group $i$, with a mean $\mu_i$ depending on the group.
If the correlation between the measurements is nonnegative, then the above model is equivalent to a mixed nested two-way ANOVA model with a fixed group effect and a random individual effect: $$y_{ijk} = \mu_i + \gamma_{ij} + \epsilon_{ijk}$$ where $\gamma_{ij} \sim {\cal N}(0, \sigma_b^2)$ is the random effect for individual $j$ in group $i$ and $\epsilon_{ijk} \sim {\cal N}(0,\sigma^2)$ are the residual terms. This can be interpreted by saying that $\mu_i$ is the theoretical mean of group $i$ and $\mu_i + \gamma_{ij}$ is the theoretical mean of individual $j$ in group $i$.
If the design is balanced (same number of individual per group and same number of repeated measurements per individual), there is a least-squares approach providing an exact confidence interval about your parameter of interest $\mu_1-\mu_2$, and this interval is exactly the same as the "Student" confidence interval obtained from the two samples $\bar y_{1j\bullet}$ and $\bar y_{2j\bullet}$ of the individual means.
