250VAC is the regulatory/safety rating for the part, which means that if you use it in an application where the steady-state voltage across it is at or less than 250VAC, it's considered 'safe' by those safety certification bodies and won't require any special investigation by the safety people when it comes time to certify the end-product. (Sometimes also known as a recognized component - the funky backwards UR mark is on this part, which is the UL Recognized Component mark.)
The capacitor is a class-Y2 device which means the part is impulse tested (i.e. for a very short time) at 5000VDC. This doesn't mean the part can operate at 5000VDC, but it's a measure of the strength of the dielectric material. Y1 is "stronger" (tested at 8000VDC).
300V is the manufacturer's absolute maximum voltage for the part. If you exceed 300V, there's no guarantee of performance or survival.
I would look for 277 or 312V Y-capacitors - they should only be marginally larger (if at all) and would eliminate any doubts about suitability.
The answers already here are correct. I just wanted to add that since you are using a Full-Wave Rectifier (4 diodes), your ripple voltage is determined by:
$$ V_{ripple}=\frac{V_{peak}}{2fCR}$$
or equivalently,
$$ V_{ripple}=\frac{I_{load}}{2fC}$$
Now, the thing here is that you want to keep the ripple voltage 'small.' A good number is within 10% of \$V_{peak}\$. From your post, I see \$V_{peak}=13.8{\mathrm V}\$
If you do the math, in order to keep the ripple voltage, say, 10% of \$V_{peak}\$, the maximum current you can draw from the rectifier is :
$$ I_{load}=2(1.38{\mathrm V})(50{\mathrm{Hz}})(4700\mu{\mathrm F})$$
$$ I_{load}=0.65{\mathrm A}$$
That is the maximum current you want to draw if you want keep the ripple voltage at a maximum of \$V_{ripple}=1.38{\mathrm V}\$ or 10% of the peak voltage.
I saw in one of your comments that you were concerned about damaging components. You won't damage them as long as they are rated to handle the voltage/current they are being provided with. The capacitor should be fine (since it is 50V), but they may be a lot of current running through the circuit at startup (when the capacitor is fully discharged). So you want to make sure your diodes are rated to handle that much current, and also check the reverse voltage ratings for those diodes.
The reverse voltage across those diodes are theoretically the same as the peak voltage for a full wave rectifier with four diodes (in your case the diodes should handle more than the 13.8 volts you are getting at the output).
Best Answer
Simply put, Y capacitors are intended for filtering common mode between line and ground/PE. X capacitors are intended for filtering across the line: L1-L2 or L-N. This is described in IEC60384-14 which is harmonised with UL 60384 and EN 60384.
You clearly do not have a Y capacitor application if you're not connecting to chassis/ground/PE.
The peak pulse voltage you would expect to encounter dictates whether you would use an X1 or X2. For up to 2500V, you can use X2; above that and I believe specified up to 4kV you would need X1.
You mentioned that you plan to obtain UL and CE-Mark for the end product. I suggest you work with your lab or certifying agent on this detail since it doesn't sound like you are using the capacitor in its intended application (line to line).