Q:

Why can't I apply for some jobs?

There are some jobs on the website that aren't marked as "Applied" in the jobs list, but the "Apply" button in the job description is not availble for them; there is only the "Save" button.
I wonder why this can be happening?
For example: https://stackoverflow.com/jobs/202114/front-end-software-developer-memorial-sloan-kettering

A:

While many jobs have applications that you can fill out without leaving Stack Overflow, some jobs have applications that are hosted outside of Stack Overflow, and all of those have an external link.
In this case, the external link is https://ad.doubleclick.net/ddm/clk/410749410;217026690;y?http://careers.mskcc.org/job/8555701/front-end-software-engineer-digital-computational-pathology-new-york-ny/. Since ad.doubleclick.net is usually used for clicking on advertisements, an ad blocker is probably blocking the entire button from appearing.
I spent about 15 minutes searching for other job postings that use external links, but I couldn't find any, so I can't tell you how common it is for the links to go through ad.doubleclick.net. If this was a choice by the company who made the application by providing that link, then there's not really anything Stack Overflow Jobs can do about it. However, if SO has it set up so that all external job application links go through ad.doubleclick.net before getting to the final destination, then perhaps a feature request is in order to have that changed.
This could be a viable feature request since Stack Overflow has already said "we don’t care if our users use ad blockers on Stack Overflow,", in part, because "impressions served to [users] will only annoy them." Since Stack Overflow cares about not annoying their users, and disappearing job buttons could be considered pretty annoying when using a pretty standard (in many cases considered necessary)1 tool like ad-block, they would probably be willing to listen to a feature request regarding changing how the off-site job application buttons are served. 
Of course, not caring about ad blockers is not equivalent to saying they're willing to spend the dev time to make every little part of the site work around them (If I recall correctly, there was a time that Ad Block Plus was removing all of the social media buttons, and the response was something like "If you want to use those buttons, just turn off ad block"). If this is the fault of the company providing a link that is blocked, then I can't see Stack Overflow taking any action, but if this is done by Stack Overflow, then they might be willing to make a change to how the button is rendered.
1There are many reasons already on the web about why using adblockers can help with security, such as when an ad is hacked to include malware or redirect your page, so some companies go to great lengths to make sure their website is fully functional below an ad blocker since they expect so many users to use them.

