How can I fit distribution for data which "almost fits"? I have a sample for events occurring at certain continuous distances (kilometers), let's suppose emergency calls to hospitals. I have 200k observations, coming from 500 hospitals for an entire month. So, when I plot a histogram for my  distances vector, it  looks to me like a Beta distribution with determined parameters, but actually it's not. Below we have these histograms with respectively 50 and 20k bins:
 
Yes, the second histograms shows some spikes and pits, and this is maybe my biggest problem to fit, because every goodness-of-fit test gives me extremely low p-values, (and I have already tried to use smaller samples with the same shape).
When I use Probability Plot my results seems to be "kinda good", but not perfect.

I've been reading about method of moments and kernel density estimation, in order to understand if I can use them to fit my data to a pdf (even if not from the theoretical ones), but first I would like to know some things:

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*Is there anything I could or I should do about my data before estimation? Is window smoothing a valid alternative?

*Should I give up trying to fit it into beta? Is there a way to tell something like "this is beta, but with error margins"? What could I say in the paper to support my decision?

*If my answer isn't in any of the alternatives above, what should I be reading right now to advance?

I need to fit it to make some conclusions about fuel and maintenance of the cars.

The answer by @Ben, which you've already accepted, is great.
I'll just add that a Beta distribution has bounded support (assumes an upper maximum), whereas you're dealing with distances, which don't naturally lend themselves to such an assumption. Moreover, the QQ plot indicates a potential uncertainty in the tail of the fitted distribution.
Therefore, I recommend to also try to fit to your datasets:

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*a Gamma distribution (perhaps constrained with shape parameter $k = \alpha = 2$),

*a Weibull distribution (perhaps constrained with shape parameter $k = 2$, a case which is equivalent to a Rayleigh distribution).

https://en.wikipedia.org/wiki/Gamma_distribution
https://en.wikipedia.org/wiki/Weibull_distribution
https://en.wikipedia.org/wiki/Rayleigh_distribution (equivalent to Weibull with shape parameter $k = 2$)
