I have used my Pi3 with several USB powered portable hard drives. A single drive should work with any decent 5V/2A+ power supply and good USB cables on both sides (power to Pi3 and HDD cable).
I'm using my Western Digital Passport 320GB USB 2.0 hard drive right now to backup some files.
Power input to my Pi3 is showing 5.26V with current varying in the 780mA-1.25A range. Power at the USB port is showing 4.88V and current is varying between 300-610mA (with brief spikes in current going a bit higher). I'm using a 5.3V/2.4A PSU.
I have even successfully spun up and mounted two external USB powered hard drives, but I could only use one at a time (trying to use both at once would result in intermittent drive disconnects).
The Raspberry Pi3 can supply up to 1.2A to the USB ports (all 4 ports combined). So one HDD should be easy, but two is not going to be reliable.
Besides the HDD, I also have a Logitech Unifying dongle for my wireless keyboard and mouse, plus a 16GB USB flash drive and my digital USB multimeter. WiFi is on and connected to my router. Bluetooth is on, but not in use at the moment. I'm running the latest Raspbian Jessie Pixel, fully updated. Chromium browser is open with 5 tabs, and Audacious is playing music from the flash drive. HDMI is connected to a 1080p monitor, and the analog audio out is connected to some powered speakers.
The copy operation has finished and power input is now showing 5.29V @ 700(ish)mA, while the USB port meter is indicating 5.05V @ 280mA (drive is idle, but has not spun down).
So yes, if you have a good power supply and good USB cables you can definitely run a USB hard drive from the Pi3 USB ports. But just one... if you need more than that, you'll need either a powered hub, or powered hard drives.
FYI: The USB 2.0 spec is 500mA per port, so 4 ports should be able to supply 2A. The Raspberry Pi3 is well below spec at only 1.2A.
My mind is like a browser. 27 tabs are open, 9 aren't responding,
lots of pop-ups...and where is that annoying music coming from?