Mavericks requires 8GB's or more of storage space.
Click your Apple menu icon top left in your screen. From the drop down menu click About This Mac > More Info > Storage
Make sure there's at least 15% free disk space.
If it's not a storage space issue ...
- Disable anti virus software if installed
- Make sure the firewall is turned off in System Preferences > Security & Privacy > Firewall
- Make sure Gatekeeper is enabled
Open System Preferences > Security & Privacy then select the General tab.
Make sure either Mac App Store or Mac App Store and identified developers is selected. If that area is grayed out, click the padlock icon to proceed.