I am thinking of building a model predicting a ratio $a/b$, where $a \le b$ and $a > 0$ and $b > 0$. So, the ratio would be between $0$ and $1$.

I could use linear regression, although it doesn't naturally limit to 0..1.  I have no reason to believe the relationship is linear, but of course it is often used anyway, as a simple first model.

I could use a logistic regression, although it is normally used to predict the probability of a two-state outcome, not to predict a continuous value from the range 0..1.

Knowing nothing more, would you use linear regression, logistic regression, or hidden option c?

You should choose "hidden option c", where c is beta regression.  This is a type of regression model that is appropriate when the response variable is distributed as Beta.  You can think of it as analogous to a generalized linear model.  It's exactly what you are looking for.  There is a package in R called betareg which deals with this.  I don't know if you use R, but even if you don't you could read the 'vignettes' anyway, they will give you general information about the topic in addition to how to implement it in R (which you wouldn't need in that case).  



Edit (much later):  Let me make a quick clarification.  I interpret the question as being about the ratio of two, positive, real values.  If so, (and they are distributed as Gammas) that is a Beta distribution.  However, if $a$ is a count of 'successes' out of a known total, $b$, of 'trials', then this would be a count proportion $a/b$, not a continuous proportion, and you should use binomial GLM (e.g., logistic regression). For how to do it in R, see e.g. How to do logistic regression in R when outcome is fractional (a ratio of two counts)?

Another possibility is to use linear regression if the ratios can be transformed so as to meet the assumptions of a standard linear model, although I would not be optimistic about that actually working.  
