Yes, it's possible (in the US only) with the standard Authorize.net integration.

First, you need to have an Authorize.net account that is set up with a processor and your Authorize.net account must have the eCheck.Net feature activated.

Then, you must install the Authorize.net module in your Odoo instance. Afterwards, go to the Accounting app > Configuration > Payment Acquirers, and make sure you configure your Authorize.net acquirer with the API login ID and API Transacaction Key that you got from Authorize. You must also ensure that this acquirer is "Published" in the button at the top of the form view.

[EDIT March 11 2017]: Depending on your Odoo version, you also might have to remove the following line in Accounting/Configuration/Payment acquirer/Authorize/Configuration tab/Form Button Template/Architecture field: 

if you want to allow both the options of credit card payment or ACH payment.

[EDIT Feb 7 2019]: Alternatively, you can replace it with 

if you want to force the credit card option only or the ACH option only.

Once this is done, your ecommerce checkout page will display the option of paying with Authorize.net and the customer will have the option to pay with ACH by providing his bank account information.

[EDIT Sep 3 2021]: Starting in Odoo 15, ACH payments via Authorize.net are possible with an embedded form (like all other acquirers) and support one-shot flows (online quote, online invoice, ecommerce checkout) as well as recurring subscriptions flows with tokenization. There is no more redirect and no more Form Button Template. From v15 on, ACH via Authorize.net is no more an undocumented feature but officially maintained.

[EDIT Apr 27 2024]: Starting in Odoo 17, ACH in Stripe is supported (like any other payment method supported in Stripe thanks to the support in Odoo 17 of Stripe Elements).