Times of sunrise and sunset are calculated with a "duration of daylight" formula. For the purposes of that formula, Earth's surface is considered a sphere, atmospheric refraction is disregarded and the centre of the Sun is considered as the whole Sun.
The corrections, from this theoretical model to the real world have been known for centuries. They do not change the "yes" answer to your question. They would change the date at which "simultaneous sunrise-sunset" occurs for a specific location in the Arctic (or Antarctic), but the idea remains the same: there may be a day in the year (more often two) when the top limb of the Sun's visible disk barely touches (is tangent) to the observer's local horizon. One day before (in Spring) the noon Sun is clearly below the horizon (by a few arcminutes); one day after, the top of the Sun's disk "peeks" above the horizon for a few seconds.
P is the local "Polar Angle" of the Sun's hour circle (relative to the observer's meridian). This angle is zero when the Sun crosses the meridian, the moment when the Sun's apparent altitude (relative to the horizon) is highest. It is also called "transit" (or upper transit, to distinguish from the lower transit, at the time of midnight Sun). Since the Sun appears to go one full turn (360 degrees) in 24 hours, the rate is 15 degrees per hour.
DEC is the declination of the Sun (its apparent angular distance from the celestial equator). It goes from +23.4368 degrees to -23.4368 degrees.
LAT is the latitude of the observer (0 = equator, +90 = North pole, +66.5632 is the Arctic circle)
The "length of day" formula is
cos(P) = - tan(LAT)*tan(DEC)
P is the Polar angle of the Sun at sunrise (or sunset). P/15 is the time (in hours) to go from sunrise to transit, or the time to go from transit to sunset.
2P = total length of day (time period when the Sun is above the local horizon)
IF you find the conditions for P to be exactly zero, then the length of day (on that particular day, at that particular location) will be exactly zero. Conclusion: sunrise and sunset are at the exact same time.
As it turns out, if the values of LAT and DEC are complementary (they add up to 90 degrees), AND their signs are opposite (one is North while the other is South) then the multiplication of -tan(LAT)*tan(DEC) will be exactly 1. A cosine of 1 corresponds to an angle of zero.
If they are of same sign (for example, both North), then the multiplication gives -1. The angle P is 180 degrees and the "length of day" 2P = 360 -- which corresponds to 24 hours: the Sun "sets" for zero seconds.
In theory, if you are at latitude of 66.5632 N (Arctic circle) on the day of December solstice (DEC = 23.4368 S), the Sun's centre will "touch" the local horizon at exactly local solar noon (at the moment of transit) and the duration of tangency will be zero (as soon as it touches the horizon, it starts to go down again).
In practice, because of air refraction (roughly 1 degree at the horizon) and the apparent radius of the Sun (a little more than a quarter of a degree), you may have to go 75 nautical miles further north (LAT = approx 67.8 N) to witness this... err... day of length zero.