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Even though, there is a command line interface on linux (which has start, stop and, soon, pause options), there seems to be no equivalent feature on windows nor on mac. This feature should be easily ...
I haven't actually tried this with a Raspberry Pi but I have with x86/x86-64 based machines and it worked.
You might be able to install dropbox onto a network attached Raspberry Pi and plug in an external hard drive to the Pi. (For external hard drives you might need a powered usb hub if the EHD requires power [the Pi won't have enough juice to power it, unless you boost its usb output current from about 600mA to 1.2A - which I read might damage your Pi {oh and configure /etc/fstab to auto-mount the drive if you restart the pi} ] ).
Use putty to control/setup your headless Rpi-NAS server remotely.
Map a network drive between your pc to the network attached storage (install and setup samba on the Pi for this [and change password to make it secure]).
Oh, almost forgot. You can use Chron to schedule tasks and maybe use that with some scripting language like BASH or Python.
Chron: (Like task scheduler but for GNU/Linux)
https://www.raspberrypi.org/documentation/linux/usage/cron.md
If you get the Pi to work, let me know. Here are some very helpful guides:
Raspbian: (The lite version takes up way less space.)
https://www.raspberrypi.org/downloads/raspbian/
Putty SSH client: (for remote access from windows)
Dropbox for Raspbian:
https://www.raspberrypi.org/magpi/dropbox-raspberry-pi/
/etc/fstab guide:
https://www.cyberciti.biz/faq/linux-finding-using-uuids-to-update-fstab/
passwd: make it secure
https://www.raspberrypi.org/documentation/linux/usage/users.md
Samba:
BASH:
http://ryanstutorials.net/bash-scripting-tutorial/
Python: (make sure you select the correct Python version before reading the wrong tutorial)