I have OS X 10.9.5, and my Calendar (version 7.0) will not let me change the "Days start at:" and "Days end at:" in Calendar general preferences. It will allow me to change the times inside the actual preference box (in the scroll menu), and it will stay to the specifications I set it at. However, it does not change on the actual calendar: every day in the weekly view starts at 1am and ends at midnight, no matter which time it is set to.
The "Show: … hours at a time" option works still; it's only the "Days start at:" and "Days end at:" options that don't work.
Best Answer
There appears to be a misunderstanding about what the "Day starts at" and "Day ends at" settings do in Mavericks. The underlying idea is that these are your work hours.
Back when it was still called iCal, this caused portions of days outside that range to be shaded, to show they were off-hours. Now, however, the only effect these settings have (from what I have been able to find) is that they are used to determine which address is the starting location, for estimating travel time to an event. During work hours, your "work" address is used, and during other hours, your "home" address is used.
This is overridden by Calendar if there is an event (with a location) within the three hours prior to an event's start time, in which case the starting location is assumed to be the prior event's location. If you don't have work and/or home addresses set in your contact card in Contacts, Calendar will use your computer's current location (if available).
Regardless of the "Day starts at" and "Day ends at" settings (and even prior to Mavericks), the entire day will always be shown.