Supporting geolocation in Apex seems simple enough - the Geolocation javascript API (more info here) contains all you need, and it is supported by all modern browsers.
However, getting this to work properly in Apex took me a lot longer than I expected.
The solution below does not use geolocation.watchPosition, which was the javascript function call I started out with. Using that seemed logical, since we want to "watch" the position and update it in the database only when neccessary. However, getting that function to reliably (and not too often) update the page items turned out to be impossible (for me anyway...).
Also you don't want to update the coordinates too often, since that creates a lot of calls to the database. But you do want to update it often enough to be accurate. You need a document cookie for that, as shown below (more info here).
So now I use geolocation.getCurrentPosition, setInterval, some cookies, and a package/procedure to read the cookies and set the page items. This works like a charm.

Note 1: I put the javascript in the page itself for this demo. You will probably want to include it in the page template instead, so that updates can occur on every page. In that case, remove all "document.getElementById" lines from the various functions.

Note 2: The get_geo_cookie page process needs to be included on all pages that require an up-to-date geolocation. Of course in that case the javascript needs to be available on those pages as well (e.g. in the page template).

Note 3: You need to allow your browser to (always) share your location with this website.

Note 4: For sites that use authentication, on the login page you should put some code to reset the latitude/longitude document cookies.
Put this in Page -> JavaScript -> Execute when Page Loads:

document.cookie="geo_latitude=9999";
document.cookie="geo_longitude=9999";
document.cookie="geo_error=");

Note 5: This feature is available only in secure contexts (HTTPS), in some or all supporting browsers.

Page

Identification
	Page ID	1060
	Name	Apex - HTML / CSS / JavaScript - Geolocation
Javascript
	Function and Global Variable Declaration	// Geolocation: update the cookie with current coordinates function geo_success(position) { var geo_latitude = position.coords.latitude; var geo_longitude = position.coords.longitude; document.cookie = "geo_latitude=" + geo_latitude; document.cookie = "geo_longitude="+ geo_longitude; document.cookie = "geo_error="; document.getElementById('P1060_LATITUDE' ).value = geo_latitude; document.getElementById('P1060_LONGITUDE').value = geo_longitude; document.getElementById('P1060_ERROR' ).value = ''; } // Geolocation: handle errors function geo_error(err) { console.warn('GEO_ERROR(' + err.code + '): ' + err.message); document.cookie = "geo_error=" + "(" + err.code + "): " + err.message; $s('P1060_ERROR', err.code + ': ' + err.message); document.getElementById('P1060_ERROR').value = err.code + ': ' + err.message; }; // Geolocation: options (high accuracy) var geo_options_high = { enableHighAccuracy: true, /* Is a Boolean that indicates the application would like to receive the best possible results. If true and if the device is able to provide a more accurate position, it will do so. Note that this can result in slower response times or increased power consumption (with a GPS chip on a mobile device for example). On the other hand, if false, the device can take the liberty to save resources by responding more quickly and/or using less power. Default: false. */ timeout: 20000, /* Is a positive long value representing the maximum length of time (in milliseconds) the device is allowed to take in order to return a position. The default value is Infinity, meaning that getCurrentPosition() won't return until the position is available. */ maximumAge: 0 /* Is a positive long value indicating the maximum age in milliseconds of a possible cached position that is acceptable to return. If set to 0, it means that the device cannot use a cached position and must attempt to retrieve the real current position. If set to Infinity the device must return a cached position regardless of its age. */ }; // Geolocation: run getCurrentPosition (high accuracy) function geo_location_int_high() { navigator.geolocation.getCurrentPosition(geo_success, geo_error, geo_options_high); document.getElementById('P1060_ACCURACY').value = 'HIGH'; } // Geolocation: options (low accuracy) var geo_options_low = { enableHighAccuracy: false, timeout: 20000, maximumAge: 60000 }; // Geolocation: run getCurrentPosition (low accuracy) function geo_location_int_low() { navigator.geolocation.getCurrentPosition(geo_success, geo_error, geo_options_low); document.getElementById('P1060_ACCURACY').value = 'LOW'; }
	Execute when Page Loads	// Geolocation: update the coordinates once with low accuracy, then once with high accuracy, and then repeat every X milliseconds with high accuracy geo_location_int_low(); geo_location_int_high(); setInterval("geo_location_int_high()", 30000);

Item

Identification
	Sequence	10
	Name	P1060_LATITUDE
	Display As	Text Field
User Interface
	Region	Geolocation
Advanced
	Warn on Unsaved Changes	Page Default

Item

Identification
	Sequence	20
	Name	P1060_LONGITUDE
	Display As	Text Field
User Interface
	Region	Geolocation
Advanced
	Warn on Unsaved Changes	Page Default

Item

Identification
	Sequence	30
	Name	P1060_ACCURACY
	Display As	Text Field
User Interface
	Region	Geolocation
Advanced
	Warn on Unsaved Changes	Page Default

Item

Identification
	Sequence	100
	Name	P1060_ERROR
	Display As	Textarea
User Interface
	Region	Geolocation
Advanced
	Warn on Unsaved Changes	Page Default

Process

Name
	Button Name	get_geo_cookie
	Type	PL/SQL anonymous block
Process Point
	Sequence	10
	Process Point	On Load - Before Header
	Run Process	Once Per Page Visit (default)
Source
	Process	begin ted_p1060.get_geo_cookie; end;

Package

Package
	Name	TED_P1060
	Source	package ted_p1060 is procedure get_geo_cookie; end ted_p1060;

Package Body

Package Body
	Name	TED_P1060
	Source	package body ted_p1060 is procedure get_geo_cookie is v_cookie owa_cookie.cookie; begin apex_util.set_session_state('P1060_LATITUDE', ''); apex_util.set_session_state('P1060_LONGITUDE', ''); apex_util.set_session_state('P1060_ERROR', ''); v_cookie := owa_cookie.get('geo_latitude'); apex_util.set_session_state('P1060_LATITUDE', v_cookie.vals(1)); v_cookie := owa_cookie.get('geo_longitude'); apex_util.set_session_state('P1060_LONGITUDE', v_cookie.vals(1)); v_cookie := owa_cookie.get('geo_error'); apex_util.set_session_state('P1060_ERROR', v_cookie.vals(1)); exception when no_data_found then -- no cookie available null; end get_geo_cookie; end ted_p1060;