There are currently (I can't speak for the future) three related services that when used in combination can help you, although most people probably don't know that they can be used without a mobile phone:
- The Gmail Phone (this isn't hardware; it's a web app you can use from your Gmail account)
- Google Voice (this is what actually gives you a phone number that can be associated with the Gmail Phone; without Google Voice, you can still use it, but you won't have a phone number that people can call from regular phones)
- Google Hangouts (for video and audio chat, if you don't need to call phone numbers; it looks like they recently added Gmail Phone like functionality to Google Hangouts)
There are some requirements, however:
- Google Voice requires that you associate your account with either a landline or a mobile phone number upon registration. You can remove it later (since it's not necessary for your purposes), but I believe you need it to start with. Two Google Voice accounts can be associated with one landline, and one Google Voice account can be associated with a mobile phone. It sounds like a Skype number will currently work, too, since you told me you tested it out and it worked.
- You have to install one or two plug-ins.
- For the Gmail Phone, you may have to use the Chrome web browser or something besides Firefox (currently; Firefox has worked in the past).
Here are the instructions:
- Install the Google Chrome and Google Hangouts.
- go to voice.google.com (or google.com/voice); on my browser, at the moment, the former takes me to the new Google Voice site and the latter takes me to the legacy Google Voice site (these instructions are for the new site).
- register for an account/number
- During registration, select the desktop computer option for signing up (on a desktop computer)
- Verify your phone number that you're using to sign up with by phone—not by text, unless that's an option for you.
- You may want to choose a local number (or a number local to any landline phones without long distance you may want to use; e.g. like a church or work phone). The reason for this is outlined further on.
- Follow the prompts.
- Go to Gmail (with the same Google account you used for Google Voice; a different one won't work)
- Make sure chat is turned on.
- Click the phone icon in the lower left corner. (You may or may not have to press the blue link that says 'Make a call' a ways above it after you press the phone button.)
- A box will pop up where you can type a name or phone number to dial.
- Try to call someone. If you don't have it installed, it may prompt you to install the Google Talk plugin (I thought the Google Hangouts plug-in replaced that years ago, but it looks like they have a new plug-in with that name.)
Your Google Voice phone number should automatically be associated with your Gmail Phone. However, if it's not, go to voice.google.com. Click the main menu button in the upper left corner (it doesn't say main menu unless you hover over it). Then click on Legacy Google Voice. Then click on the settings dropdown button near the upper right corner (and click on settings). You should see the associated phones with your account, which includes the Gmail Phone, although it calls it Google Chat, currently (even though it's not a physical phone). Make sure it's checked. Click on the edit button underneath it. If it says, 'show advanced settings', then show them. Configure your phone as desired. Save. Look at the other tabs for more settings (including for your voicemail PIN, if you want to use that).
You may also try Google Hangouts, as it seems to allow the same functionality as the Gmail Phone, now, but that seems new.
Anyway, these are free services as long as you're making calls as long as you're not calling regular phones. They are also free if you're calling regular phones in the USA and Canada. You have to buy credit to call outside the USA and Canada, however.
These services are also very useful if you don't have your own phone or a computer, but you do have access to a local phone you can visit, even if it can't naturally call long-distance. If you go to such a phone, you can call your Google Voice number, press *, enter your PIN, press 2, and then you can dial anywhere in the USA and Canada for free. You don't need to associate this phone with your account (you can do it from any phone). You can also check your voicemail, and make international calls using credits you've purchased. It should be noted that if the phone you use doesn't have long-distance, you do need a local Google Voice phone number in order to call it to do all this stuff (so, choose your Google Voice number wisely, and research which numbers are local from where).
You can always get more than one account, though. As far as I know, there's nothing against it.
You can also send and receive text messages with your Google Voice number (at voice.google.com). You can have them forwarded to your email. Replying to the emails will reply to the texts.
Skype is an alternative to Google Voice; it offers phone numbers as you discovered, and may end up easier to use, but it doesn't have as many free services (and requires a software installation instead of browser plug-ins).
Anyway, I don't have a cell phone, and I haven't always had a landline or a tablet either. So, I've relied on Google Voice quite a bit. Even without setting up the Gmail Phone, it's still nice since you can text and receive voice mail. You can also block callers, forward your number to multiple phones and stuff like that. I've only occasionally used the Gmail Phone for incoming calls, but I have done it. Now, I just forward them to the house's landline so I can have my own personal voicemail and phone number and still ring on the same phone as everyone else.
Just so you know, in case you ever do get a tablet, Skype is probably more convenient for making calls on tablets, but Google does seem to be working on improving things currently. There is at least one Android app that historically has been able to integrate with Google Voice (with a tablet but without a phone), but I don't know if it still works (it was called Groove IP Light). I managed to use it on my Kindle Fire (second edition; not HD) with the right kind of microphone headset (since that kind of Kindle Fire doesn't have an obvious microphone, although it does work with the right kind of microphone headset) for a while, at least. I don't think the newer versions of the program (or even the older ones, anymore) work on my device, or even on my Kindle Fire HD8 (6th edition), but they might on a more Google-oriented tablet (Amazon and Google don't often mix well); no guarantees, though. Skype works fine on my Kindle Fire HD8, but it probably drains the battery more than without it and might add some bloat.