How can I use time stamps for classification? My data consists of entries when an event is True, namely, when a train crossing is down. So I will have entries within a day like so (just examples):

start,end,interval-length

10:43:02,10:46:02,180,
10:49:06,10:51:06,120

The state is of course True for these entries, and the times not logged throughout a day are when the state is False, aka traversable. (however I have a strong feeling I should explicitly pad these times in, so that I have full 24-hour cycles logged)

The data presented is a time series, based on which I wish to make predictions for coming days. I want to be able to ask "What are the chances that the crossing is down at time $t$, of day $d$?" (the second argument is important because I will have 7 different models, one for each day of the week due to the differing train traffic schedules in a week), taking into consideration the previously observed times.

This of course needs to take into account the fact that when the events happen, they have a certain length (as shown by my data). This is not inherently a classification problem because I am also interested in a confidence of the prediction, which is not as simple as taking a built model's performance parameters. Assuming though that this could be a promising way to tackle the problem, how can I build a model so that the times later on in a day don't wrongfully bias it? 

Consider the simple example of some dataset, where days are represented as indexing numbers 1-7. An unsupervised classification algorithm would be biased to guess more towards saturday and sunday, because they are higher numbers (and in this case the solution is to have 7 different columns, each with boolean values instead - but I can't do anything similar to this in my case). I hope this example clears up my question a bit.
Not knowing your data in detail, my intuition is that you could go for dummy (one hot) encoding. You could split each day in (say) 10 min. intervals $x$ (144 columns) and attach labels (up/down) $y$. Each time interval $x$ would be one dummy encoded column (true/false).

The model would be a binary classification (logistic) like:

$$ y = \beta_0 + \beta_1 x_1 + ... + \beta_n x_n + u .$$

In case you have enough days, this could be a reasonable predictive model. It is likely, that some of the dummy encoded $x$ are irrelevant. So it could be worth to check a Lasso model, where some (not so helpful $x$) are shrunken or automatically excluded from the model. 

https://web.stanford.edu/~hastie/glmnet/glmnet_alpha.html

Based on the same logic, you could also check boosting, e.g. lightGBM to make the classification.