  We report results from long-term (approximately 1240 days) X-ray (0.3-8.0
keV) monitoring of the ultraluminous X-ray source NGC 5408 X-1 with the
Swift/X-Ray Telescope. Here we expand on earlier work by Strohmayer (2009) who
used only a part of the present data set. Our primary results are: (1) the
discovery of sharp, quasi-periodic, energy-independent dips in the X-ray
intensity that recur on average every 243 days, (2) the detection of an
energy-dependent (variability amplitude decreases with increasing energy),
quasi-sinusoidal X-ray modulation with a period of 112.6+-4 days the amplitude
of which weakens during the second half of the light curve and (3) spectral
evidence for an increase in photoelectric absorption during the last continuous
segment of the data. We interpret the X-ray modulations within the context of
binary motion in analogy to that seen in high-inclination accreting X-ray
binaries. If correct, this implies that NGC 5408 X-1 is in a binary with an
orbital period of 243+-23 days, in contrast to the 115.5 day quasi-sinusoidal
period previously reported by Strohmayer (2009). We discuss the overall X-ray
modulation within the framework of accretion via Roche lobe overflow of the
donor star. In addition, if the X-ray modulation is caused by vertically
structured obscuring material in the accretion disk, this would imply a high
value for the inclination of the orbit. A comparison with estimates from
accreting X-ray binaries suggests an inclination > 70 degrees. We note that, in
principle, a precessing accretion disk could also produce the observed X-ray
modulations.
