The calendar point is possible. But the special relativity point, not so much.
While "now" can vary according to the time it takes to get from A to B, all observers can work out what "now is according to the reference frame of the distant stars". On a mere interplanetary scale, this idea is amazingly precise. We might not agree on how much time passed (this has already mattered for GPS satellites), but for all practical purposes "now" is perfectly meaningful.
Yeah, looks like it would work out fine. Since synchronous communication breaks down pretty quickly even on merely interplanetary levels, the remaining use for calendars and timekeeping is basically event planning. For that difference in reference points and interval duration is not that important, as long as you can compensate for that, making events happen at a time you expect in a place you expect. We're doing that fine already, communicating with our deep space probes.
While "now" can vary according to the time it takes to get from A to B, all observers can work out what "now is according to the reference frame of the distant stars". On a mere interplanetary scale, this idea is amazingly precise. We might not agree on how much time passed (this has already mattered for GPS satellites), but for all practical purposes "now" is perfectly meaningful.